Questions tagged [congruences-geometry]
For questions about congruent geometric figures, e.g., determining whether one figure is congruent to another.
101
questions
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24 views
Does there exist an efficient way to determine if two $n$-simplices in $\mathbb{R}^n$ are congruent?
Given two $n$-simplices in $\mathbb{R}^n$, does there exist an algorithm to determine if they are congruent which has a time complexity of the form $O(n^c)$ where $c$ is a constant?
I'm assuming that ...
5
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1answer
104 views
Prove ABC,A'B'C' are congruent:D is on BC,D' is on B'C', $\angle BAD \angle CAD= \angle B'A'D' \angle C'A'D', AB=A'B', AC=A'C', AD=A'D'$
In $\triangle ABC$ and $\triangle A'B'C'$, $D$ is a point on line segment $BC$
and $D'$ is a point on line segment $B'C'$. $\frac{\angle BAD}{\angle CAD}=\frac{\angle B'A'D'}{\angle C'A'D'}$, $AB=A'B'$...
1
vote
1answer
65 views
Triangle and Median related question
Let BD be a median in triangle ABC. The points E and F divide the
median BD in three equal parts, such that BE = EF = FD. If AB = 1 and
AF = AD, find the length of the line segment CE.
I have tried ...
1
vote
4answers
101 views
Find the measure of $\angle G$ in a triangle
In the following figure, $\Delta$ADB, $\Delta$PCB and $\Delta$EFG are right triangles.
$PB=AE$,$AC=CB$.
Question: What is the value of $\angle G$?
I figured out $\angle A = \angle CPB$ , $EF=AC$. ...
1
vote
1answer
45 views
Geometry find the angle
Good day, excuse me, I am looking at congruence themes in Geometry, and I am trying to solve this problem, but I am not sure how to proceed. The coloured numbers is what I have so far (although now ...
1
vote
3answers
69 views
Similarity by SSA (side-side-angle) in obtuse triangles
Source: Challenge and Thrills of Pre-College Mathematics, Page 74, Problem 54:
"In two obtuse triangles, an acute angle of the one is equal to an angle of the other, and sides about the other ...
2
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1answer
86 views
Finding side lengths of a trapezoid given the distance between its diagonal intersection and midpoint of a diagonal
The Question
My friend recently gave me a problem that I was interested in, but could not completely solve. I've already constructed a diagram that roughly represents the problem, and it's below.
...
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3answers
33 views
Let $ABCD$ be a square. Points $A,B,G$ are collinear. $AC$ and $DG$ meet at $E$ , and $DG$ and $BC$ meet at $F$.
Let $ABCD$ be a square. Points $A,B,G$ are collinear. $AC$ and $DG$ meet at $E$ , and $DG$ and $BC$ meet at $F$. If $DE = 15$ cm, $EF = 9$ cm, find $FG$.
What I Tried: Here is a picture :-
I have ...
0
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2answers
132 views
In the diagram, $AB$ $||$ $EF$ $||$ $DC$ . Given that $AC + BD = 250$ , $BC = 100$ and $EC + ED = 150$, find $CF$.
In the diagram, $AB$ $||$ $EF$ $||$ $DC$ . Given that $AC + BD = 250$, $BC = 100$ and $EC + ED = 150$, find $CF$.
What I Tried: Here is the diagram :-
I have assigned variables for the different ...
0
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2answers
49 views
$\Delta ABC$, $AC = 2BC$ and $\angle C = 90^\circ$ and $D$ is the foot of the altitude from $C$ onto $AB$. Find $AE : EC$ .
$\Delta ABC$, $AC = 2BC$ and $\angle C = 90^\circ$ and $D$ is the foot of the altitude from $C$ onto $AB$. A circle with diameter $AD$ intersects the segment $AC$ at $E$. Find $AE : EC$.
What I Tried:...
4
votes
2answers
211 views
In $\Delta ABC,$ side $AC$ and the perpendicular bisector of $BC$ meet at $D$, where $BD$ bisects $\angle ABC$.
In $\Delta ABC,$ side $AC$ and the perpendicular bisector of $BC$ meet at $D$, where $BD$ bisects $\angle ABC$. If $CD = 7$ and $[\Delta ABD] = a\sqrt{5}$ , find $a$ .
What I Tried: Here is a picture:...
1
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1answer
53 views
Let $O$ be the centre of the circumcircle of $\Delta ABC$, $P$ and $Q$ be the midpoint of $AO$ and $BC$, respectively.
Let $O$ be the centre of the circumcircle of $\Delta ABC$, $P$ and $Q$ be the midpoint of $AO$ and $BC$, respectively. Suppose $\angle CBA = 4\angle OPQ$ and $\angle ACB = 6\angle OPQ$ . FiNd $\angle ...
2
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0answers
21 views
Planar convex sets whose self-intersections are similar to themselves
Let $S\subset \mathbb{R}^2$ be a bounded convex set. For which $S$ can we take $S'$ congruent to $S$ (i.e., the image of $S$ under an isometry of the Euclidean plane) such that $S\cap S'$ is similar ...
0
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3answers
62 views
In $\Delta ABC$, $AB = 14, BC = 16, AC = 26$. $M$ is the midpoint of $BC$ and $D$ is the point on $BC$ such that $AD$ bisects $\angle BAC$.
In $\Delta ABC$, $AB = 14, BC = 16, AC = 26$. $M$ is the midpoint of $BC$ and $D$ is the point on $BC$ such that $AD$ bisects $\angle BAC$. Let $P$ be the foot of the perpendicular from $B$ to $AD$ . ...
1
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1answer
92 views
In $\triangle ABC, AB = 28, BC = 21$ and $CA = 14$. Points $D$ and $E$ are on $AB$ with $AD = 7$ and $\angle ACD = \angle BCE$
In $\triangle ABC, AB = 28, BC = 21$ and $CA = 14$. Points $D$ and $E$ are on $AB$ with $AD = 7$ and $\angle ACD = \angle BCE$. Find $BE$.
What I Tried: Here is a picture :-
I know the side-lengths ...
4
votes
1answer
165 views
In $\Delta ABC$, $AB:AC = 4:3$ and $M$ is the midpoint of $BC$ . $E$ is a point on $AB$ and $F$ is a point on $AC$ such that $AE:AF = 2:1$
In $\Delta ABC$, $AB:AC = 4:3$ and $M$ is the midpoint of $BC$ . $E$ is a point on $AB$ and $F$ is a point on $AC$ such that $AE:AF = 2:1$. Also $EF$ and $AM$ intersect at $G$ with $GF = 36$ cm, $GE = ...
0
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1answer
27 views
How to prove that a triangle is uniquely determined by an angle, its opposite side and its perpendicular height.
I am trying to solve a homework problem and as part of this problem, I need to show that a certain situation is impossible. I have the following situation. Given some line, I have $2$ points $H$ and $...
0
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1answer
105 views
When extending the sides $AB,BC,CA$ of $\Delta ABC$ to $B',C',A'$ respectively, such that $AB' = 2AB$ , $CC' = 2BC$ , $AA' = 3CA$ .
When extending the sides $AB,BC,CA$ of $\Delta ABC$ to $B',C',A'$ respectively, such that $AB' = 2AB$ , $CC' = 2BC$ , $AA' = 3CA$ . If $[\Delta ABC] = 1$, find $[\Delta A'B'C']$ .
What I Tried: Here ...
1
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2answers
77 views
In $\Delta ABC$, angle bisector of $\angle ABC$ and median on side $BC$ intersect perpendicularly
In $\Delta ABC$, $BE$ is the angle bisector of $\angle ABC$, $AD$ is the median on side $BC$. $AD$ intersects $BE$ at $O$ perpendicularly. If $AD = BE = 4$, find the lengths of each side of $\Delta ...
2
votes
1answer
64 views
In $\Delta ABC$, $AC = BC$ and $\angle C = 120^\circ$. $M$ is on side $AC$ and $N$ is on side $BC$ .
In $\Delta ABC$, $AC = BC$ and $\angle C = 120^\circ$. $M$ is on side $AC$ and $N$ is on side $BC$ such that $\angle BAN = 50^\circ$ and $\angle ABM = 60^\circ$. Find $\angle NMB$ .
What I Tried: ...
1
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2answers
42 views
In right $\Delta ABC$, $\angle C = 90^\circ$. $E$ is on $BC$ such that $AC = BE$. $D$ is on $AB$ such that $DE \perp BC$ .
In right $\Delta ABC$, $\angle C = 90^\circ$. $E$ is on $BC$ such that $AC = BE$. $D$ is on $AB$ such that $DE \perp BC$ . Given that $DE + BC = 1$ and $BD = \frac{1}{2}$, find $\angle B$.
What I ...
1
vote
1answer
73 views
In an isosceles right $\Delta ABC$, $\angle B = 90^\circ$. AD is the median on BC. Let $AB = BC = a$.
In an isosceles right $\Delta ABC$, $\angle B = 90^\circ$. AD is the median on BC. Let $AB = BC = a$. If $BE \perp AD$, intersecting $AC$ at $E$, and $EF \perp BC$ at $F$, find $EF$ in terms of $a$.
...
-1
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1answer
80 views
$ABCD$ is a rectangle of area $210$ cm$^2$. $L$ is a mid-point of $CD$ . $P,Q$ trisect $AB$ . $AC$ cuts $LP,LQ$ at $M,N$ respectively.
$ABCD$ is a rectangle of area $210$ cm$^2$. $L$ is a mid-point of $CD$ . $P,Q$ trisect $AB$ . $AC$ cuts $LP,LQ$ at $M,N$ respectively. Find $[\Delta LMN]$
What I Tried: For a geometry Problem, it's ...
2
votes
4answers
65 views
$ABCD$ is a square. $E$ is the midpoint of $CB$, $AF$ is drawn perpendicular to $DE$. If the side of the square is $2016$ cm , find $BF$.
$ABCD$ is a square. $E$ is the midpoint of $CB$, $AF$ is drawn perpendicular to $DE$. If the side of the square is $2016$ cm , find $BF$.
What I Tried: Here is a picture,
I used a really peculiar ...
1
vote
1answer
150 views
$ABCD$ is a square. $E$ and $F$ are points respectively on $BC$ and $CD$ such that $\angle EAF = 45^\circ$.
$ABCD$ is a square. $E$ and $F$ are points respectively on $BC$ and $CD$ such that $\angle EAF = 45^\circ$. $AE$ and $AF$ cut the diagonal $BD$ at $P,Q$ respectively. Find $\frac{[\Delta AEF]}{[\Delta ...
1
vote
1answer
63 views
Let $ABCD$ be a rectangle where $\Delta PAB$ is isosceles. The radius of the circles are $3$,$4$,$3$ cm respectively.
Let $ABCD$ be a rectangle where $\Delta PAB$ is isosceles. The radius of each of the smaller circles is $3$ cm and the radius of the bigger circle is $4$ cm. Find the length and breadth of the ...
2
votes
1answer
86 views
$ABCD$ is a square, with $K$ and $L$ are points on $BC$ and $DC$ respectively . If $AM \perp LK$ AND $\angle AKM = \angle AKB$ , Find $\angle LAK$ .
Here is a diagram if needed :-
What I Tried :- I did angle-chasing , considered $\angle MKA = \angle AKB = x$ and then I did not get any information for some other angles , so I considered $\angle ...
0
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0answers
39 views
is still there a contradiction in the proof by contradiction of the proposition 6 of Elements using this triangle?
The proposition 6 says:
If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.
And the proof of Euclid in his Elements basically is (for ...
1
vote
2answers
232 views
Is the criterion SAA of congruent triangles valid?
I think the fundamental criterions for triangles congruence are:
SAS (Side-Angle-Side)
ASA (Angle-Side-Angle)
SSS (Side-Side-Side)
But some proofs like this one: https://www.youtube.com/watch?v=...
0
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2answers
42 views
Does congruent triangles apply to this question?
Two identical rods $BA$ and $CA$ are hinged at $A$. When $BC = 8\ \textrm{cm}$, $\angle BAC = 30^\circ$ and when $BC = 4\ \textrm{cm}$, $\angle BAC = \alpha$. Show that $$\cos\alpha = \frac{6+\sqrt 3}{...
0
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1answer
26 views
Area of a rectangle using congruency
I don't understand the lecturer, solving the question, says that the rectangles are congruent each other, so the result can be obtained by proportioning them to each other. However, AFAIK two ...
0
votes
1answer
32 views
Problem on an isosceles right triangle, involving similarity and congruence
Given that $ABC$ is an isosceles right angled triangle with angle $\widehat{ACB}=90$ degrees. $D$ is the midpoint of $BC$, $CE$ is perpendicular to $AD$, intersecting $AB$ and $AD$ at $E$ and $F$ ...
1
vote
2answers
51 views
Congruency and parallelism in a triangle
For the following example, one of the solutions says $\frac{6-x}{6} = \frac{x}{18}$ if one of the side of the square is called as $x$. I don't understand how the proportion and the equality are ...
1
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2answers
76 views
minimum number of congruent rectangles
consider a 6*6 square which is dissected into 9 rectangles by lines parallel to its sides such that all the rectangles have integral sides.the question is - what is the minimum number of congruent ...
1
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1answer
54 views
Proof that a spherical triangle is congruent with its dual
I'm currently studying spherical geometry and ran into an exercise problem that I'm having trouble understanding.
The book first defines a dual spherical triangle $\triangle^* ABC$ of original $\...
1
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1answer
133 views
Triangle congruence when the longest sides, the largest angles, and one of the other sides are congruent?
Are two scalene triangles (a scalene triangle is one with no two equal
sides) congruent if their longest sides, largest angles, and shortest sides are congruent?
I believe the case described above is ...
3
votes
2answers
79 views
Avoiding Circular Reasoning: How to Define Congruent Shapes
I apologize for being overly verbose here, but the question I want to know is at the very bottom. I am going to be honest and say I have no idea how to axiomatically handle congruence in geometry and ...
3
votes
3answers
176 views
Congruence of angles axiom
In my geometry book, the following statement is provided as axiom of congruence.
Let $ABC$ and $A'B'C'$ be two triangles. If $AB \equiv A'B'$, $AC \equiv A'C'$ and $ \angle BAC \equiv B'A'C'$ then $ \...
2
votes
2answers
969 views
When we have two line segments $AB$ and $CD$, what does $AB=CD$ mean?
Suppose that we have two line segments, AB and CD. We know that they have the same length.
I know that $\overline{AB}=\overline{CD}$ means $AB$ is identical to CD (aka. They are the same lines), and ...
2
votes
3answers
303 views
Proving two triangles are congruent
The two circles shown are identical and pass through each other's centre. The line $AC$ passes through the centres of both circles.
I would like to prove that triangles $ABD$ and $BCE$ are congruent ...
3
votes
1answer
232 views
Olympiad Geometry | Homothety
Let $C$ be a point on line segment $AB$, and construct the circles with diameters $AB$ and $AC$. Let $M$ be the midpoint of $BC,$ and let $MD$ be a tangent to the smaller circle (with $D$ on the ...
3
votes
1answer
249 views
Proving that DEF is also an equilateral triangle
The question gives that ABC is an equilateral triangle, AD = BE = CF and ∠DAB = ∠EBC.
I tried using only cases of triangle congruences to prove it, but I'm not sure if I did it correctly. Thus, take a ...
2
votes
2answers
54 views
How to solve this congruence of triangles geometrically?
Find x.
Solution with trigonometry:
First there are the missing angles ...
Superior = $180 - 13 - 32 = 135$
Bottom = $180 - 13 - 24 = 143$
By the breast theorem we have to
$$...
-2
votes
2answers
69 views
Prove that ED=EF [closed]
In the diagram below $AD=DC$ and $AE=EB$ and both triangles $AEB$ and $ADC$ are right angel triangles and $M$ is the midpoint of $BC$ also $MD=MF$.
Prove that $ED=EF$
2
votes
4answers
226 views
Find the length x such that the two distances in the triangle are the same
I have been working on the following problem
Statement
Assume you have a right angle triangle $\Delta ABC$ with cateti $a$, $b$ and hypotenuse $c = \sqrt{a^2 + b^2}$. Find or construct a point $D$ ...
1
vote
1answer
47 views
If ratio of sides of two triangles is constant then the triangles have the same angles
If $\triangle ABC$ and $\triangle A'B'C'$ are a pair of triangles such that
$$\dfrac{|AB|}{|A'B'|}=\dfrac{|BC|}{|B'C'|}=\dfrac{|AC|}{|A'C'|}$$
then
$$\triangle ABC \sim \triangle A'B'C'$$
I have ...
0
votes
1answer
47 views
Can anyone help me to find what is the similarity between these two triangles Just check the picture)? [closed]
Can anybody show me how does the red triangle (bigger triangle) is similar to the green triangle (smaller triangle). I need to know how they are similar so that I can do a proportional between them.
2
votes
3answers
100 views
Compute $m ( \angle ACD $).
Let $\triangle ABC $ s.t $m (\angle A)=100^{°}, m (\angle B)=20^{°} $. Let $D\in Int (\triangle ABC) $ s.t. $m (\angle BAD)=30^{°} $ and $[BD $ is the bisector of $\angle B $.
Compute $m ( \angle ACD ...
1
vote
0answers
247 views
Spherical triangles and congruence criteria
I read from different sources that the usual criteria of congruence of triangles work for spherical
triangles. However it seems to me that there is a counterexample. Consider two poles on the sphere A ...
1
vote
1answer
93 views
In $\triangle ABC$, $D$ is an exterior point such that $AC = CD$ and $CE$ is parallel to $AF$. Find the area of $ABDF$.
In $\triangle ABC$, $CB$ is extended upto $D$ so that $AC$ = $CD$. An angle $\angle DCE$ is drawn at point $C$ so that is equal to $\angle CAB$ and $AB$ meets $CE$ at $I$.$E$ is such an external point ...
