Question

Solve the problems below. Please answer with completely simplified exact value(s) or expression(s).

a)
In ΔABC, AC = BC, CD⊥AB with D∈AB, AB = 4 in, and CD = 3 in. Find AC.

b)
Given: ΔABC, AB = BC = AC = a. Find: The area of ΔABC

Answer 1

a) √7

Explanation:

Answer 2

a) AC =
`√(13)`

b) Area =
`(\sqrt 3)/(4)` ×
`a^(2)`

Explanation:

a) From question,

AC = BC, CD⊥AB

Now in ΔCAD and ΔCBD

AC=BC, ∠A = ∠B and AD=BD (because in isosceles triangle perpendicular bisects the side).

then, from SAS potulates

ΔCAD≅ΔCBD

So,

AD =
`(AB)/(2)` =
`(4)/(2)` = 2 in

From Pythagorean theorem in ΔADC

`AC^(2)` =
`AD^(2)` +
`CD^(2)`

`AC^(2)` =
`2^(2)` +
`3^(2)`

`AC^(2)` = 4 + 9 = 13

AC =
`√(13)`

b) In given ΔABC,

AB = BC = AC = a, means ΔABC is a equilateral triangle.

So, area of equilateral triangle is

Area =
`(\sqrt 3)/(4)` ×
`side^(2)`

side = a

then,

Area =
`(\sqrt 3)/(4)` ×
`a^(2)`