Question

in triangle abc measure angle A + measure angle C is equal to measure angle B and ac ratio a b is equal to 25 ratio 24 if BC is equal to 21 find the area of triangle ABC

Answer 1

Answer:

The area of the triangle ABC is 756.

Step-by-step explanation:

It is provided that in a triangle ABC,

• m∠A + m∠C = m∠B
• $\frac{AC}{AB} =\frac{25}{24}$
• BC = 21

Sum of all angles of a triangle is 180°.

Then,

m∠A + m∠C + m∠B = 180°

m∠B + m∠B = 180°

2 × m∠B = 180°

m∠B = 90°

Thus, the triangle ABC is a right angled triangle.

Using the Pythagoras theorem determine the side AB:

$AC^{2}=AB^{2}+BC^{2}\\(\frac{25}{24}AB)^{2}= AB^{2}+21^{2}\\\frac{625}{576}AB^{2}= AB^{2}+441\\\frac{625}{576}AB^{2}- AB^{2}=441\\\frac{49}{576}AB^{2}=441\\AB^{2}=\frac{441\times576}{49}\\ =5184\\AB=\sqrt{5184}\\ =72$

The area of the triangle ABC is:

$Area=\frac{1}{2} \times Base\times Height\\=\frac{1}{2} \times BC\times\ AB\\=\frac{1}{2} \times 21\times\ 72\\=756$

Thus, the area of the triangle ABC is 756.