# maths

in triangle abc angle b = 2angle c.d is a point ob bc such that ad bisects angle bac and ab = cd.prove that angle bac = 72

1. 👍 0
2. 👎 0
3. 👁 205
1. make a sketch.
let angle C = x , then angle B = 2x
also, let angle BAD = y = angle CAD , (Angle A was bisected by AD)
then in triangle ABC , 2x + 2y + x = 180
y = (180-3x)/2 = 90 - 3x/2

by exterior angle theorem,
angle BDA = x+y and angle CDA = 2x+y
let AB = m = DC , (given)

by the sine law:
m/sin(x+y) = AD/sin 2x and m/siny = AD/sinx
m = ADsin(x+y)/sin 2x and m = ADsiny/sinx

sin(x+y)/sin2x = siny/sinx
sin(x + 90-3x/2)/(2sinxcosx) = siny/sinx
sin(90 - x/2)/(2cosx) = siny , after dividing by sinx
cross-multiply
sin(90-x/2) = 2sinxcosx
sin(90-x/2) = sin 2x
so 90 - x/2 = 2x
times 2
180 - x = 4x
180=5x
x=36
then y = 90-3(36)/2 = 36
and angle A = 2y = 72°

1. 👍 0
2. 👎 0
2. There is a wonderful field of math lying ahead from this problem.
notice that cos 36° = .809016.. which is 1/2 of the golden ratio of (1+√5)/2 or 1.61803....

Look up the "pentagon"
Draw the central triangles, your triangle is one of these.

1. 👍 0
2. 👎 0

## Similar Questions

1. ### Math

The area of triangle ABC is divided into four equal parts by segments parallel to the base BC. DC is 36 cm. A is the top point of the triangle. What is the ratio of the dimensions of triangle ADE to the dimensions of triangle ABC?

asked by Jeff on May 21, 2020
2. ### Precalculus

In most geometry courses, we learn that there's no such thing as "SSA Congruence". That is, if we have triangles ABC and DEF such that AB = DE, BC = EF, and angle A = angle D, then we cannot deduce that ABC and DEF are congruent.

asked by John on February 24, 2014
3. ### maths-urgently needed

In right angled triangle ABC angle B=90 degree and AB= root 34 unit. The coordinates of point B and Care (4,2)and (-1,y) respectively. If area of triangle ABC = 17 sq.units, then find value of y.

asked by Anonymous on February 12, 2013
4. ### geometry

1. Triangle ABC has a 63.0-degree angle at B, and side AC is 13.6 cm long. What is the diameter of the circle circumscribed about ABC? 2. And: Given any triangle ABC, with sides a, b, and c opposite angles A, B, and C,

asked by Esi on April 12, 2010
5. ### Math

1. Name a pair of complementary angles. (1 point) 1 and 4 1 and 6 3 and 4 4 and 5 2. If m1=37°, what is m4? (1 point) 53° 43° 37° 27° 3. If m1 = 40°, what is m5? (1 point) 50° 40° 35° 25° Use the figure to answer

asked by YRN DJ on December 17, 2015
1. ### geometry

Isosceles $\triangle{ABC}$ has a right angle at $C$. Point $P$ is inside $\triangle{ABC}$, such that $PA=11$, $PB=7$, and $PC=6$. Legs $\overline{AC}$ and $\overline{BC}$ have length $s=\sqrt{a+b\sqrt{2}}$, where $a$ and $b$ are

asked by William on May 28, 2017
2. ### Math

Triangle ABC has angles measuring 137°, 15°, and 28°. Classify the triangle by its angle measures. right triangle acute triangle obtuse triangle multiangular triangle I thinks it's c

asked by Connexus Girl on December 20, 2017
3. ### Geometry

using a ruler and a pair of compasses only. construct triangle.ABC in which angle BAC=45 °.Then |AB|=7cm.then |AC|=9cm.locate P into triangle ABC which is 5cm from A and B equidistant from B and C. find PB.

asked by latif on January 19, 2020
4. ### geometry

The coordinates of point A on triangle ABC are (2, -4). What would be the image of point A after triangle ABC is reflected over the y-axis andthen rotated about the origin 180 degree?

asked by Piggy on May 5, 2012
5. ### math

Triangle $ABC$ is a right triangle with right angle at $A$. Suppose $\overline{AX}$ is an altitude of the triangle, $\overline{AY}$ is an angle bisector of the triangle, and $\overline{AZ}$ is a median of the triangle, and \$\angle

asked by InfaRed on November 10, 2016
6. ### Math "Need Help Asap"!

Geometry Unit Practice Sheet Multiple Choice Use the figure to answer questions 1–3. 1. Name a pair of complementary angles. (1 point) 1 and 4 1 and 6 3 and 4 4 and 5 2. If m1=37°, what is m4? (1 point) 53° 43° 37° 27° 3.

asked by Markest Lee on December 17, 2015