Answer:
(a) 26.63°, 26.63°, 126.73°
(b) 25.92 cm
Step-by-step explanation:
The altitude of an isosceles triangle divides the triangle into two congruent right triangles. Each right triangle has a base equal to half the base of the isosceles triangle and a hypotenuse equal to the length of the congruent sides of the isosceles triangle.
The base angle of the isosceles triangle is equal to the angle opposite the height in one of the right triangles formed by the altitude. To find the measure of this angle, we can use the sine trigonometric ratio.
[tex]\boxed{\begin{array}{l}\underline{\textsf{Sine trigonometric ratio}}\\\\\sf \sin(\theta)=\dfrac{O}{H}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{O is the side opposite the angle.}\\\phantom{ww}\bullet\;\textsf{H is the hypotenuse (the side opposite the right angle).}\end{array}}[/tex]
In this case, the side opposite the angle is 6.5 cm, and the hypotenuse measures 14.5 cm. Therefore:
[tex]\sin\theta=\dfrac{6.5}{14.5}\\\\\\\theta=\sin^{-1}\left(\dfrac{6.5}{14.5}\right)\\\\\\\theta=26.6331187...^{\circ}\\\\\\\theta=26.63^{\circ} \sf \; (2\;d.p.)[/tex]
So, the congruent bases angles of the isosceles triangle each measure 26.63°.
Since the base angles are congruent in an isosceles triangle, and the sum of the three angles of any triangle is 180°, the measure of the other angle (apex) can be calculated as follows:
[tex]180^{\circ}-2\theta\\\\180^{\circ}-2(26.6331187...^{\circ}\\\\180^{\circ}-53.266237498...^{\circ}\\\\126.7337625^{\circ}\\\\126.73^{\circ}[/tex]
Therefore, the congruent base angles of the isosceles triangle measure 26.63° each, and the apex angle measures 126.73°.
The base of the isosceles triangle is twice the base of one of the right triangles with height 6.5 cm and hypotenuse 14.5 cm.
To find the base (b) of the right triangle, we can use the Pythagorean Theorem:
[tex]6.5^2+b^2=14.5^2\\\\42.25+b^2=210.25\\\\b^2=168\\\\b=\sqrt{168}[/tex]
As the base of the isosceles triangle is twice the base of the right triangle, then:
[tex]\textsf{Base of isosceles triangle} =2b\\\\\textsf{Base of isosceles triangle} = 2\sqrt{168}\\\\\textsf{Base of isosceles triangle} = 25.9229627936...\\\\\textsf{Base of isosceles triangle}=25.92\; \sf cm[/tex]
So, the base of the isosceles triangle measures 25.92 cm.
