Question

Lateral side of an isosceles triangle is 15 cm and the altitude is 8 cm. what is the radius of the circumscribed circle

Answer 1

There is no such thing as a “radius of a triangle”.

There are two radii associated with a triangle - the in radius and circumradius.

The in radius is the radius of the circle having all the three sides of the triangle as tangents.

The formula for the inradius is r=A/8 where A is the area of the triangle and s is the semi-perimeter.

It is given that one side of an isosceles triangle (presumably the non-congruent side) is 15 cm and the altitude (presumably from the vertex common to the congruent sides) is 8 cm.

⇒  The area of the triangle is A=12×15×8=60 cm²

The altitude bisects the non-congruent sides and is perpendicular to it.

⇒ The length of each congruent side is (8²+(152)²)−−−−−−−−−−−√=481√2.

⇒ The semi-perimeter is s=15+481√2 cm.

⇒ The inradius is r=As=60×215+481√=12015+481√=3.24924 cm.

The circumradius is the radius of the circle passing through all the three vertices of triangle.

The formula for the circumradius is R=abc/4A, where a,b and c are the lengths of the sides and A is the area of the triangle.

⇒R=15×481√/2×481√/2 / 4×60=481/64= 7.515625  cm.

Answer 2

It is given that

one side of an isosceles triangle is 15 cm and the altitude is 8 cm.

⇒The area of the triangle is A = $\frac{1}{2} \times 15\times 8$

                                                  = $\frac{120}{2}$

                                                   = $60cm^{2}$

Now, The altitude bisects the non-congruent side and is perpendicular to it.

⇒ The length of each congruent side = $\sqrt {8^{2} + (\frac{15 }{2}  )^{2}}$

                                                             $=\sqrt{64+\frac{225}{4} }$

                                                             $=\sqrt{\frac{256+225}{4} }$

                                                             $=\sqrt{\frac{481}{4} }$

                                                             $=\frac{\sqrt{481} }{2}$

Now the formula for finding the radius of the circumscribed circle is:

$R=\frac{a \times b  \times c }{4 \times A}$

Where a,b,c are the length's of sides and A is the area of triangle

$\therefore R=\frac{15 \times \frac{\sqrt{481} }{2}  \times \frac{\sqrt{481} }{2} }{4 \times 60}$

                          $=\frac{\frac{15 \times 481}{4} }{240}$

                          $=\frac{1803.75}{240} =7.5156$ cm

Therefore R = 7.5156 cm

Please ignore $\hat{A}$ in between formula's and calculations