Question Video: Finding the Perimeter of Triangle Drawn inside Another Triangle Whose Vertices Touch the Circle Mathematics

In a circle of center 𝑂, 𝐴𝐡 = 35 cm, 𝐢𝐡 = 25 cm, and 𝐴𝐢 = 40 cm. Given that line segment 𝑂𝐷 βŠ₯ line segment 𝐡𝐢 and line segment 𝑂𝐸 βŠ₯ line segment 𝐴𝐢, find the perimeter of △𝐢𝐷𝐸.

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Video Transcript

In a circle of center 𝑂, 𝐴𝐡 is equal to 35 centimeters, 𝐢𝐡 is equal to 25 centimeters, and 𝐴𝐢 is equal to 40 centimeters. Given that line segment 𝑂𝐷 is perpendicular to line segment 𝐡𝐢 and line segment 𝑂𝐸 is perpendicular to line segment 𝐴𝐢, find the perimeter of triangle 𝐢𝐷𝐸.

We are given in the question the length of the three sides of the triangle 𝐢𝐡𝐴. We know that 𝐴𝐡 is 35 centimeters, 𝐢𝐡 is 25 centimeters, and 𝐴𝐢 is 40 centimeters. We have been asked to calculate the perimeter of triangle 𝐢𝐷𝐸. We will do this by firstly proving that triangles 𝐢𝐡𝐴 and 𝐢𝐷𝐸 are similar using the chord bisector theorem. We notice from the diagram that the line segments 𝑂𝐸 and 𝑂𝐷 both pass through 𝑂 and meet the chords 𝐴𝐢 and 𝐢𝐡 at right angles.

The chord bisector theorem states that if we have a circle with center 𝑂 containing a chord 𝐡𝐢, then the straight line that passes through 𝑂 and is perpendicular to 𝐡𝐢 also bisects 𝐡𝐢. In our diagram, this means that the length of 𝐴𝐸 is equal to the length 𝐸𝐢 and the length 𝐢𝐷 is equal to the length 𝐷𝐡.

It is also clear from the diagram that 𝐴𝐢 is equal to two multiplied by 𝐸𝐢 and 𝐢𝐡 is equal to two multiplied by 𝐢𝐷. As the two triangles 𝐢𝐡𝐴 and 𝐢𝐷𝐸 also share the angle 𝐢, we have two corresponding sides in proportion and the angle between the two sides is congruent. This proves that the two triangles are similar. And in fact triangle 𝐢𝐡𝐴 is larger than triangle 𝐢𝐷𝐸 by a scale factor of two, as the lengths of the corresponding sides are twice as long. Side 𝐴𝐢 is equal to two multiplied by side 𝐸𝐢, 𝐢𝐡 is equal to two 𝐢𝐷, and 𝐴𝐡 is equal to two multiplied by 𝐸𝐷.

We can calculate the perimeter of triangle 𝐢𝐡𝐴 by adding 40, 35, and 25. This is equal to 100 centimeters. The perimeter of triangle 𝐢𝐷𝐸 will therefore be equal to half of this. This is equal to 50 centimeters.

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