Asked by Anonymous
ABC is rightangled triangle. AD is the bisector of angle BAC. Angle DAC=15 degrees. X=CD. Find X
Answers
Answered by
mathhelper
Assuming angle B - 90°, then angle A = 30° and angle C = 60°
You have the 30-60-80 triangle whose sides are 1 , √3, 2 respectively
In the right triangle ABD, angle ADB = 75° and BD = 1-x
cos 75 = (1-x)/√3
I will further assume you want "exact" values, so we need cos 75°
cos75 = cos(45+30)
= cos45cos30 - sin45sin30
= (√2/2)(√3/2) - (√2/2)(1/2)
= (√6 - √2)/4
(1-x)/√3 = (√6 - √2)/4
√18 - √6 = 4 - 4x
4x = 4 + √6 - 3√2
x = (4 + √6 - 3√2)/4
You have the 30-60-80 triangle whose sides are 1 , √3, 2 respectively
In the right triangle ABD, angle ADB = 75° and BD = 1-x
cos 75 = (1-x)/√3
I will further assume you want "exact" values, so we need cos 75°
cos75 = cos(45+30)
= cos45cos30 - sin45sin30
= (√2/2)(√3/2) - (√2/2)(1/2)
= (√6 - √2)/4
(1-x)/√3 = (√6 - √2)/4
√18 - √6 = 4 - 4x
4x = 4 + √6 - 3√2
x = (4 + √6 - 3√2)/4
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