2
votes
3answers
63 views

Finding an area of a triangle inside of a triangle, given certain areas of other triangles, and area ratios.

I'm studying for the Waterloo Math Contest (Galois, Gr. 10) taking place in April of 2015 and I am preparing by looking at previous problems and solving them. This is question 4(c) on the 2010 Galois ...
1
vote
2answers
36 views

How to find length of the sides of a triangle given the ratio of the sines of the sides?

Consider $\triangle ABC$. Let $\dfrac{\sin A}{\sin B} = \dfrac56$ and $\dfrac{\sin B}{\sin C} = \dfrac45$. Find $\dfrac{\vert AC\vert\cdot \vert AB\vert}{\vert BC\vert}$. If there is no definite ...
1
vote
1answer
32 views

Finding the area of a triangle in terms of the radius of the excircle

Prove that the area of a triangle $ABC$ is $$\frac12 (b + c - a)r$$ where $r$ radius of the excircle opposite to $A$ and the rest of the symbols have their usual meaning. I started off with the ...
4
votes
2answers
26 views

formula for number triangles

Hi, I have a triangle starting from $0$ and going up by one on the bottom row until there are $r$ items on the bottom row and there are $r$ rows a number is formed by adding the two numbers towards ...
0
votes
1answer
42 views

To prove inequality for two similar triangles $ABC$ and $A_1B_1C_1$ given that $A_1B_1C_1$ is inscribed in $ABC$

Consider a triangle $ABC$. A directly similar triangle $A_1B_1C_1$ is inscribed in the triangle $ABC$ such that $A_1,\;B_1\;,C_1$ are the interior points of the sides $AC,\;AB\;and\;BC$ respectively. ...
1
vote
1answer
28 views

Find Coordinates on a track

Charlie and Alexandra are running around a circular track with radius 60 meters. Charlie started at the westernmost point of the track, and, at the same time, Alexandra started at the northernmost ...
0
votes
2answers
41 views

Finding coordinates on a circle

So this problem I am have difficulty with. I think where I am going wrong is how to calculate the initial theta. Do I just use pi/2 because in the pictures it show to angle theta off the 90 degree ...
0
votes
1answer
29 views

Find the height of the dam given angles of a triangle

The top of a dam has an angle of elevation of 1.3 radians from a point on a river. Measuring the angle of elevation to the top of the dam from a point 155 feet farther downriver is 0.8 radians; assume ...
-1
votes
0answers
48 views

Find the radius if theta is less than 90 degrees

Matilda is planning a walk around the perimeter of Wedge Park, which is shaped like a circular wedge, as shown below. The walk around the park is 2.4 miles, and the park has an area of 0.25 square ...
1
vote
1answer
51 views

How to solve this geometry question?

Let ABC be an acute-angled triangle; L, M, N be the feet of perpendiculars respectively from A, B, C to the opposite sides; D, E, F be the midpoints of the sides BC, CA, AB respectively; and $I_1, ...
0
votes
1answer
15 views

How to find last pt of triangle

How to find last pt of triangle. I got (1,7) and (0.5, 4). The equations are y = 3|2x − 1| + 4 and y = −|x − 4| + 10
0
votes
1answer
41 views

Year 10 - Trigonometry

Please ignore the pencilled 4m in the diagram but I really need to know what the length of the bottom line - line DC - is. A procedure or tips on how to calculate this would be useful. Also, is the ...
3
votes
1answer
106 views

Inequality problem about sides of a triangle and the semiperimeter

Let $a,b,c$ the sides of a triangle and $s$ be the semi perimeter. Then show that $$ a^2+b^2+c^2 > \frac{36}{35}(a^2+\frac{abc}{s}) $$ I tried it doing in many ways using some ...
0
votes
1answer
33 views

An Inverse Cosine Problem

Here is my problem: $$ \sin(\cos^{-1} \frac{2}{5} ) $$ I know how to do it for the most part; I just draw a triangle with sides 2,5 and √21 and I then find the sine (opposite/hypotenuse) of the ...
0
votes
1answer
36 views

Solve an Angle-Side-Angle special case triangle if it has an obtuse angle?

I've seen this type of problem multiple times on homework, and it's confusing me like mad. The scenario: We have a triangle. It is a special case triangle, with one angle, one side, and another ...
1
vote
1answer
38 views

Find out the $\angle PRQ$

please, help me to solve this.How can I proceed.I just need help. $PQR$ is a triangle. $M$ is a point on $QR$.here,$QM=1/3RM$ , $\angle RPM=30^ \circ$ and $ \angle QPM=20^ \circ$ now,$ \angle PRQ=??$ ...
1
vote
1answer
71 views

Orthocentre of a triangle defined by three lines

Problem: If the orthocentre of the triangle formed by the lines $2x+3y-1=0$,$x+2y-1=0$,$ax+by-1=0$ is at the origin, then $(a,b)$ is given by? I would solve this by finding poins of intersection and ...
0
votes
3answers
89 views

Perpendicular lines inside and outside a circle

No trigonometry allowed. Let $\Delta ABC$ be inscribed inside a circle.Let $P$ be a point on the circle.Let $PD$ and $PE$ be perpendiculars on on $BC$ and $AC$ respectively.Let $DE$ when extended ...
1
vote
1answer
50 views

Finding ratio of external division in a triangle.

Given a $\triangle ABC$ and $P$ dividing $AB$ internally in the ratio $2:3$ $Q$ dividing $AC$ internally in the ratio $1:2$ , with $PQ$ produced and $BC$ produced intersecting in $R$ , to find the ...
3
votes
2answers
105 views

Given the length of two altitudes and one side , find the area of triangle.

Segments $BE$ and $CF$ are the altitudes in $\triangle ABC$. $E$ is on line $AC$ and $F$ is on line $AB$. $BC = 65$, $BE = 60$ and $CF = 56$. Find $A(\triangle ABC)/100$. By the Pythagorean ...
4
votes
3answers
63 views

In $\triangle ABC$, I is the incenter. Area of $\triangle IBC = 28$, area of $\triangle ICA= 30$ and area of $\triangle IAB = 26$. Find $AC^2 − AB^2$

In $\triangle ABC$, I is the incenter. Area of $\triangle IBC = 28$, area of $\triangle ICA = 30$ and area of $\triangle IAB = 26$. Find $AC^2 − AB^2$. Here is a sketch that I drew: From the given ...
2
votes
2answers
84 views

A question related to triangles , areas , ratio of areas of triangles.

I know the title is confusing but that is because of 150-character limit, if anyone of you can improve it , please do. Consider $\triangle ABC.$ Choose a point $D$ on segment $BC$ such that ...
1
vote
2answers
69 views

A problem related to circle , altitude , triangle.

Consider a $\triangle ABC.$ Draw circle $S$ such that it touches side $AB$ at $A$. This circle passes through point $C$ and intersects segment $BC$ at $E.$ If Altitude $AD ...
2
votes
2answers
156 views

In △ABC, median AM = 17, altitude AD = 15 and the circum-radius R = 10. Find BC^2

Question is as per title. Here is a sketch that I made : By Pythagorean theorem , DM is 8. Now how can I calculate BD and MC? I still haven't found a way to utilize the information that the ...
5
votes
2answers
145 views

Minimum area of a triangle

In triangle inscribed circle with radius $r = 1$ and one of it sides $a=3$. Find the minimum area of triangle? Ans = 5.4 My reasonings: $BC = a$, $AC = b$, $AB = c$ $AD=AF=x$ $FC=CE=y$ ...
2
votes
1answer
46 views

Solution for the value of an angle of a triangle ABC

Find value of angle m< DBC Where $$BD=DC=AC$$ $$2(m\langle BAC)=14(m\langle ABD)=7(m\langle BCD)$$ I tried hard but im out of ideas now, I know the answer is 20 but I want to know how, thanks ...
0
votes
2answers
42 views

Prove that this is an isoceles triangle

I'm trying to solve a problem here. It says: "Prove that a triangle is isoceles if $\large b=2a\sin\left(\frac{\beta}{2}\right)$." $B-\beta$ I've tried to prove it but I can't Can anyone help me?
0
votes
0answers
24 views

Prove that the given triangle is isoceles

I'm trying to solve a problem here. It says: "Prove that a triangle is isoceles if $\large b=2a\sin\left(\frac{B}{2}\right)$." $B-beta$ I've tried to prove it but I can't Can anyone help me?
1
vote
2answers
37 views

issues with geometry triangle

$4$ line drawn parallel to base of triangle such that they are equidistant.if the area of the most bottom part is 4 sq cm. find area of triangle? MY THOUGHTS : being weak in geometry i couldn't make ...
1
vote
1answer
103 views

How do I solve for the height of a triangle?

The basic triangle looks something like this: How do I solve for $h$? As an example, in one problem I was given $b = 45, c = 42, \angle C = 38^\circ$ I understand how $h$ divides $\triangle ABC$ ...
0
votes
2answers
80 views

Solving all possible triangles?

So we're doing oblique triangles -- Law of Sines and all that good stuff =). I have a bunch of problems that ask you to solve for "all possible triangles that satisfy the given conditions". For ...
0
votes
1answer
89 views

How to solve bearing of oblique triangle

I'm having a hard time finding the solution of the bearing given in our example. Our Example: Suppose there's a triangle with points named A,B, and C. Point A is named Bacoor. Point B is named San ...
1
vote
0answers
58 views

Area of a triangle using vectors

I have to find the area of a triangle whose vertices have coordinates O$(0,0,0)$, A$(1,-5,-7)$ and B$(10,10,5)$ I thought that perhaps I should use the dot product to find the angle between the ...
1
vote
2answers
102 views

Two parallelograms are equal in area.

I tried this question by constructing a line $PD$ therefore forming two triangles $ADP$ and QDP but couldn't establish the congruency relation between the triangles. My approach was that if I have ...
0
votes
1answer
86 views

equality of triangle inequality

$z$ and $w$ be nonzero complex numbers. How do I show that $|z+w|=|z|+|w|$ if and only if $z=sw$ for some real positive number $s$. I approached this by letting $z=a+ib$, and $w=c+id$, and kinda ...
1
vote
1answer
38 views

Simple Question on Triangles…

What times the sum of the squares of the sides of a triangle is equal to the sum of the squares of the medians of the triangle.
1
vote
1answer
34 views

A question on similar triangles…

In the given figure, AB, EF and CD are parallel lines. Given that EG = 5 cm, GC = 10 cm and DC = 18 cm, then EF = ??
0
votes
1answer
34 views

A question on equilateral triangles

A point D is on the side BC of an eqilateral triange ABC such that DC = 1/4 BC . Then AD^2 = ??? Image I drew... Options are 13 CD^2 , 9 AB^2 , 6 CD^2 , 12 BC^2 .
1
vote
4answers
103 views

Mensuration question

I recently came across a puzzling question: Two rectangles ABCD and DBEF are as shown in the figure. The area of DBEF is: Figure (hand-made): I know that through Pythagoras, we get ...
0
votes
1answer
47 views

height and bisector in a right triangle

We have right angle $ABC$ where $AB$ is hypotenuse, and angle bisector $CD$ we know that $AB=c$, $CD=u$. Express height depending on the $c, u$, and what is the condition between $c,u$ that the ...
1
vote
3answers
75 views

Area of a critical Triangle

help me to solve this this problem please: In a triangle $ABC$, $\angle BAC$ = $60\,^{\circ}$,$AB=2AC$.Point P is inside the triangle such that $PA=\sqrt{3}$,$PB=5$. What is the area of triangle $ABC ...
2
votes
3answers
127 views

Find out the angle of <ABC

Help me to solve it please.how could it be done.I tried but nothing comes out.Help me please
2
votes
1answer
120 views

In Triangle ABC , BM and CN are perpendiculars from points B and C on any line passing through A. If L is the mid-point of BC, prove that ML = NL

I found this question in my textbook and I think this question requires the use of the mid-point theorem. I even tried proving the equality using congruence but couldn't seem to make a headway. I am ...
1
vote
2answers
68 views

Proof using properties of an isosceles or right-angle triangle

Given a triangle $ABC$ with sides $AB=BC$ and angle$\angle B=100^\circ $, prove that $$a^3 + b^3 = 3a^2b$$ where $a=AB=BC$ and $b=AC$, I have tried to use simultaneously the sine and cosine rules as ...
2
votes
1answer
87 views

A question about a very peculiar triangle.

If we have a triangle where the Perimeter >0 and the Area >0 , and Area=Perimeter, what special condition must the angles of this triangle satisfy for this to happen? I've done a bit of research and ...
1
vote
0answers
120 views

Finding Areas in triangles using ratios

What theorem/theorems should be used to find the shaded area? Y and M lie on the sides Ab and Bc respectively of the triangle YMB such that AY/MI= 1/4 and O/M = 1/3. Area ABC=35 PC and QA intersect ...
1
vote
2answers
67 views

Parallelograms in triangles

if posssible, could you only give me a few theorems in order to assist me in this question. Thankyou in advance! Links to simple websites would also be appreciated. In triangle $ABC$ $F$ is midpoint ...
-1
votes
3answers
128 views

Nature of a triangle with vertices $z_1, z_2$ and $-1$ such that $|z_1|=|z_2|=1=z_1+z_2$ [closed]

If $z_1$ and $z_2$ are distinct complex number such that $|z_1|=|z_2|=1$ and $z_1+z_2=1$, then the triangle in the complex plane with $z_1,z_2$ and $-1$ as vertices must be: equilateral. right ...
1
vote
3answers
117 views

Right Triangle Trig

I need to find the measure of each angle indicated and round to the nearest tenth. I am given two sides 12 and 13 and one angle C which is 90 degrees. How do I figure this out?
1
vote
3answers
563 views

Triangle inscribed in circle, vertex at circle's center, solve for unknown angles.

$O$ is the center of the circle , $A$ and $B$ lie on the circle what are the possible values of $x$ and $y$ I found answers options , asked to mark one or more ...