For questions about properties and applications of triangles

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1answer
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Triangles, flagpoles and heights, oh my!

Here is a math question i got from school: On a horizontal plane, there are two flagpoles. One is 20m, and the other is 10m. There is a wire connected from the top of each flagpole, to the bottom of ...
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1answer
33 views

Is any property of orthocenter related in this question?

While practicing mathematics Olympiad questions , i got the below given question . Though the solution is given , I am not able to bypass certain steps ... Can anyone please explain me why angle KPA ...
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2answers
35 views

CD is height of right-angled triangle ABC, M and N are midpoints of CD and BD: prove AM⊥CN

I was having some troubles proving this: CD is the height that corresponds to the hypotenuse of right-angled triangle ABC. If M and N are midpoints of CD and BD, prove that AM is perpendicular to CN. ...
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1answer
41 views

Distance over Time and the Pythagorean Theroem

I know the Pythagorean thereom for the last part. I am not $100\%$ sure with the other parts. Here is the problem: Marty and Rediat got in a fight. They walked away from each other on seperate paths ...
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0answers
40 views

Issue with a right-angled triangle

The area of the right angle triangle is $18\text{ cm}^2$ and the ratio of its legs is $2:3$. What is the length of the hypotenuse? I assumed the lengths of two sides to be $2x$ and $3x$. I used ...
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1answer
16 views

Which of the following are the correct angle measures for angles 1 and 2 in the triangle shown below?

http://www.explorelearning.com/ELContent/gizmos/ELMath_Deliverable/ExplorationGuides/Geometry/images/EL_GEO_TriSum6.gif A. mangle1 = 43°, mangle2 = 137° B. mangle1 = 137°, mangle2 = 43° C. mangle1 ...
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1answer
51 views

Given a particular triangle that has been constructed, I want to prove that one of the angles must be $> 45$ degrees. [duplicate]

Suppose you are given an acute triangle $XYZ$ with the following properties: At $\angle XZY$, the $\angle$ bisector is drawn and extended all the way to $XY$. Lets call the point where it intersects ...
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2answers
53 views

Two triangles with two equal sides and equal area will have the third size also equal?

Consider two triangles $\triangle abc$ and $\triangle def$ such that $ab=de$ and $ac=df$.Also area of $\triangle abc$ is equal to area of $\triangle def$.Now draw $cm$ perpendicular to $ab$ and $fn$ ...
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21 views

Two questions about triangle that blocked at rectangle…

The area of the triangle is equal to the half area of the rectangle? The center point of the triangle is same as the center point of the rectangle? About 2 - if not, how do I calculate the center? ...
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2answers
38 views

Calculating the perimeter of triangle inside of a circle

In triangle $DCB$, $BC = 10$ and is also the diameter. If the area of triangle $DCB = 11$, then determine the perimeter of the triangle. I am a little stuck on this problem. I tried using the sine ...
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2answers
814 views

Triangle in Triangle

I have the lengths of three sides of an acute triangle ABC as shown below. Assume a point P on the side AB such that, if Q is the projection of P onto BC, R is the projection of Q onto CA, P becomes ...
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1answer
40 views

Will two triangles with two equal sides and equal area have same altitude

Consider two triangles ABC and DEF.AB=DE and AC=DF .Also area of triangle ABC is equal to the area of triangle DEF.If we draw an altitude (to one of the equal sides) in both triangles, is it(altitude) ...
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1answer
25 views

What is the measurement of side KL in a scalene triangle

Given: $\triangle FGH\cong \triangle JKL$ What is the measure of side 2 on triangle 2? Here is what I have. Triangle 1: Side 1: Unknown Side 2: Unknown Side 3: 12 Triangle 2: Side 1: Unknown ...
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1answer
23 views

Perimeter Of A Simple Triangle

Here in $ \triangle ABC$ $ AC=4 , DE= EF =1, \angle ABC=90^{\circ} $. The perimeter of the triangle $ \triangle ABC$ can be written as $ \sqrt {m } + n $ where $m$ and $n$ are non-negative ...
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1answer
30 views

Finding functions in Inscribed Triangle

If we have a circle of radius $R$ around center $O$ and its inscribed triangle $XYZ$ that is acute as well as scalene. $XY$ is the longest side. $XA,YB, ZC$ are the altitudes of the triangle $XYZ$. ...
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1answer
37 views

Meaning of “circumference”

I am French and I have to solve a math problem written in English. The wording is the following : " In triangle ABC, the angle bisector of angle A intersects line BC at D and the circumference of ...
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1answer
107 views

I need help with this geometry question.

Let $ABC$ be a triangle with $AB=AC$. If $D$ is the midpoint of $BC$, $E$ is the foot of the perpendicular drawn from $D$ to $AC$ and $F$ the mid-point of $DE$, prove that $AF$ is perpendicular to ...
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0answers
27 views

Triangle Section Side Lengths

Point $D$ is on side $BC$ of $\triangle ABC$, with $AB=3$, $AC=6$, and $\angle CAD = \angle DAB = 60 ^{\circ}$. What is the length of $AD$?
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27 views

Side Section Lengths in a Right Triangle

Right triangle ABC has its right angle at C. Let M and N be the midpoints of AC and BC, respectively, with AN=19 and BM=22. What is AB?
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1answer
44 views

Area of a triangle inside a larger triangle

It's been a while since I've done any geometry so I'm a bit confused by this question. We have a triangle $\triangle PQR$ whose total area is $90 \mathrm{cm}^2$. Another triangle $\triangle PTU$ is ...
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2answers
38 views

Finding Orthocenter in Coordinate Geometry

If a triangle is formed by the equations \begin{gather}2x+3y-1=0\\ ~~x+2y-1=0\\ ax+by-1=0\end{gather} and has its orthocentre at origin, then what are the values of $a$ and $b$? (Please also tell me ...
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127 views

Finding an angle between side and a segment from specified point inside an equilateral triangle

Here is the question: $\overset{\Delta}{ABC}$ is an equilateral triangle. D is a point inside triangle. $m(\widehat{BAD})=12^\circ$ $m(\widehat{DBA})=6^\circ$ $m(\widehat{ACD})=x=?$ I managed to ...
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1answer
39 views

For planar triangulation, equivalence between 4-connectedness and non existence of separating triangle.

I want to prove the following equivalence: "A planar triangulation is 4-connected if and only if it has no separating triangle." My attempts so far: $\Rightarrow$: If there is a separating ...
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2answers
28 views

Usage of law of sines

The vertex angle of an isosceles triangle is 35 degrees. The length of the base is 10 centimeters. How many centimeters are in the perimeter? I understand the problem as there are two sides with ...
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0answers
25 views

Equilateral Triangels - geometry- minimum sums

In the following figure, the triangle ABC is arbitrary and so is the point P in its interior. We construct the two equilateral triangles APE and ABD. Show that PA+PB+PC=DE+EP+PC. Conclude from here ...
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1answer
18 views

Trigonometry (non right angled triangles)

The height of a vertical tower is to be found by a surveyor. The angle of elevation of the top of the tower from a point on the horizontal ground some distance away is measured as 28.7 degrees. From ...
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1answer
53 views

Can an equilateral triangle be an isosceles triangle, too?

I've looked in a math book that an isosceles triangle has at least two congruent sides. I also know that the words "at least" mean this symbol: $\ge$, which means "is greater than or equal to" or "is ...
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2answers
184 views

How is the hypotenuse the longest side of any right triangle?

I see that the hypotenuse of a right triangle is opposite the right angle, but how is it always the longest side? I also know that it connects to endpoints of other sides. Please help me out with ...
3
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1answer
43 views

Inequality of area of two triangles

Let $ABC$ be a triangle with sides $a,b,c$ and $A_1B_1C_1$ be another triangle with sides $a+\frac{b}2$, $b+\frac{c}2$, $c+\frac{a}2$. Prove that: $$\frac94[ABC]\le[A_1B_1C_1]$$ I tried using ...
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3answers
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Maximum value of $\sin A+\sin B+\sin C$?

What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it. Can anyone help? :)
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Proving a triangle equilateral given condition $al_a^2+bl_b^2+cl_c^2=9R\Delta$

$ABC$ is a triangle, with $l_a$, $l_b$, $l_c$ as angle bisectors, $R$ as circumradius and $\Delta$ as area, such that: $$al_a^2+bl_b^2+cl_c^2=9R\Delta$$ Is it true that $ABC$ is equilateral? I am ...
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1answer
47 views

To calculate side of the Equilateral triangle

The figure is an equilateral triangle. 3 line segments , which meet at a(any) point in the triangle , are of the length 5cm, 4cm, and 3 cm as shown in the figure. Find the side of the equilateral ...
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Prove that OD is a the angle bisector of the angle BOC.

Let ABC be a non-isosceles triangle and I be the intersection of the three internal angle bisectors. Let D be a point of BC such that $ID \perp BC$ and O be a point on AD such that $IO \perp AD$. ...
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1answer
32 views

Right Triangles

Right triangle ABC has hypotenuse AC, angle CAB=30°, and BC=√2. Right triangle ACD has hypotenuse AD and angle DAC=45°. The interiors of ABC and ACD do not overlap. Find the length of the ...
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2answers
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Triangle Segments

In a triangle $ABC$, $AB=5$, $BC=16$, $AC= \sqrt{153}$, and $D$ is on segment $BC$. Compute the sum of all possible integral measures of $AD$. I've been having trouble trying to solve this problem ...
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2answers
68 views

Beautiful little geometry problem about sines

Given triangles ABC and $A_1B_1C_1$ such that $\sin A = \cos A_1, \sin B = \cos B_1, \sin C = \cos C_1$. What are the possible values for the biggest of these 6 angles? I tried some stuff like sine ...
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3answers
50 views

A strange contradictive problem

First a part of the set of same balls was arranged into an equilateral triangle, 19 balls were not used, but when the sides of this triangle were needed to be one-uped, it was a 5 ball insufficiency. ...
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2answers
38 views

Is it possible that two triangles satisfy these conditions?

Are there two triangles with equal angles and a pair of equal sides which are not congruent? If yes, please give an example.
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1answer
25 views

Prove $a^2\cos B\cos C+b^2\cos C\cos A+c^2\cos A\cos B\leq2S.$

Prove that in any triangle inequality holds: $$a^2\cos B\cos C+b^2\cos C\cos A+c^2\cos A\cos B\leq2S.$$ Is gender inequality that occurs right triangle, not an equilateral triangle. For this reason ...
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1answer
23 views

Area of a triangle.

The area of a triangle $ABC$ is $144$.Denote the midpoint of $BC$ by $P$,of $AP$ by $Q$ and of $AC$ by $R$.Calculate the area of the triangle $PQR$. I draw the picture but I do not have any idea to ...
0
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1answer
96 views

Finding Y coordinate of third triangle point when X coordinate and two other points are already known

Suppose you know the coordinates for points A and B of a triangle. We can refer to those coordinates as (Ay,Ax) and (By,Bx). Also, suppose you know the X coordinate for point C (Cx) but do not know ...
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2answers
72 views

Is HHH a congurence criteria for triangles?

I wanted to know if a triangle defined by its 3 heights is unique. I took this up as a challenge but was able to get nowhere, can anyone help me? :)
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1answer
36 views

Trigonometric ratios

I'm stuck with a problem. Given is a triangle $\Delta ABC$ with $\angle A = 35°, BC=3$ and $AC=5$. I need to find the two possible values for $\angle C$. I only managed to found one angle. I did the ...
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1answer
31 views

When is $3R\le 2h_{\max}$ true for acute triangles?

I was working on a problem recently, and it happened that it could be solved if $3R\le 2h_{\max}$ was true for all acute angled triangles. So I used GeoGebra to check it, and found that for some ...
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3answers
237 views

Drawing a Right Triangle With Legs Not Parallel to x/y Axes?

I have been presented with an interesting problem. How can I decide whether a right triangle with given side lengths can be placed (with integer coordinate vertices) on a Cartesian plane so that the ...
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2answers
49 views

Find angle and hypotenuse of right angled triangle

Find the missing side and the hypotenuse of a right triangle that has a side length of 5 cm and a perimeter of 30 cm. I'm confused. Can somebody please explain to me how to do this step by step? Not ...
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2answers
35 views

Configuration of five or more mutually equidistant points in space.

How is it proved that there is no configuration of five or more mutually equidistant points in $R^3$? Is it done by induction? I'm stuck. Help would be appreciated. Well, surely equilateral ...
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1answer
27 views

Ratio of area between similar triangles

This question has nearly no information and I've been stuck on this for quite some time. I tried drawing the median from A thru G but the 1x to 2x ratio didn't seem to help.
3
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2answers
42 views

Triangle similarity question

I've been trying to solve this question for like 40 mins straight and can't seem to get anywhere. I tried drawing a parallel to |KM| from C to |AB| but that didn't seem to help. I just can't see a ...
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0answers
27 views

How is the Uniqueness of Equilateral Tetrahedra Proved? [duplicate]

Equilateral tetrahedrons all have this property: For any two of its vertices exists a third vertex, which forms an equilateral triangle with these 2 vertices. (It doesn't necessarily have to be a ...