1) In triangle ABC, AB=AC and D is a point on line AC so that line BD bisects angle ABC. If BD=BC, what is the measure, in degrees, of angle A?
2) In a certain iscoceles right triangle, the altitude to the hypotenuse has length 4sqrt2. What is the area of the triangle?
3) In triangle ABC, AB=17, AC=*, and BC=15. Let D be the foot of the altitude from C to AB. Find the area of triangle ACD.
Please help soon, thanks!
2) In a certain iscoceles right triangle, the altitude to the hypotenuse has length 4sqrt2. What is the area of the triangle?
Area = (1/2)hypotenuse * (altidude drawn to the hypotenuse) (1)
Area (1/2) product of the leg lengths (2)
Equate (1) and (2) and let the leg lengths = L and we have
(1/2) hypotenuse * ( 4sqrt 2 ) = (1/2) L^2 multiply through by 1/2
hypotenuse * ( 4sqrt(2) = L^2
The hypotenuse = sqrt ( L^2 + L^2) = sqrt (2 L^2 ) = Lsqrt 2
L sqrt 2 * 4* sqrt 2 = L^2 divide through by L
4 sqrt 2 * sqrt 2 = L
4*2 = L
8 = L
So...the area = (1/2) L^2 = (1/2) * 8^2 = 32 units^2
1) In triangle ABC, AB=AC and D is a point on line AC so that line BD bisects angle ABC. If BD=BC, what is the measure, in degrees, of angle A?
Call the each bisected angle. a
And call angle BCA, b
And since BD = BC...then angle BDC = angle BCD = angle BCA = b
So....since AB = AC....then angle ABC = angle BCA ⇒ 2a = b ⇒ 4a = 2b
So..in triangle BDC....
a + 2b =180
a + 4a = 180
5a = 180
a = 36
And b = 2a = 72
So...in triangle ABC
Angle ABC + Angle BCA + Angle A = 180
2a + b + Angle A = 180
72 + 72 + Angle A = 180
144 + Angle A = 180
Angle A = 180 - 144 = 108°