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

# Question Video: Finding the Perimeter of Triangle Drawn inside Another Triangle Whose Vertices Touch the Circle Mathematics • Third Year of Preparatory School

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In a circle of center π, π΄π΅ = 35 cm, πΆπ΅ = 25 cm, and π΄πΆ = 40 cm. Given that line segment ππ· β₯ line segment π΅πΆ and line segment ππΈ β₯ line segment π΄πΆ, find the perimeter of β³πΆπ·πΈ.

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### Video Transcript

In a circle of center π, π΄π΅ is equal to 35 centimeters, πΆπ΅ is equal to 25 centimeters, and π΄πΆ is equal to 40 centimeters. Given that line segment ππ· is perpendicular to line segment π΅πΆ and line segment ππΈ is perpendicular to line segment π΄πΆ, find the perimeter of triangle πΆπ·πΈ.

We are given in the question the length of the three sides of the triangle πΆπ΅π΄. We know that π΄π΅ is 35 centimeters, πΆπ΅ is 25 centimeters, and π΄πΆ is 40 centimeters. We have been asked to calculate the perimeter of triangle πΆπ·πΈ. We will do this by firstly proving that triangles πΆπ΅π΄ and πΆπ·πΈ are similar using the chord bisector theorem. We notice from the diagram that the line segments ππΈ and ππ· both pass through π and meet the chords π΄πΆ and πΆπ΅ at right angles.

The chord bisector theorem states that if we have a circle with center π containing a chord π΅πΆ, then the straight line that passes through π and is perpendicular to π΅πΆ also bisects π΅πΆ. In our diagram, this means that the length of π΄πΈ is equal to the length πΈπΆ and the length πΆπ· is equal to the length π·π΅.

It is also clear from the diagram that π΄πΆ is equal to two multiplied by πΈπΆ and πΆπ΅ is equal to two multiplied by πΆπ·. As the two triangles πΆπ΅π΄ and πΆπ·πΈ also share the angle πΆ, 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 πΆπ΅π΄ is larger than triangle πΆπ·πΈ by a scale factor of two, as the lengths of the corresponding sides are twice as long. Side π΄πΆ is equal to two multiplied by side πΈπΆ, πΆπ΅ is equal to two πΆπ·, and π΄π΅ is equal to two multiplied by πΈπ·.

We can calculate the perimeter of triangle πΆπ΅π΄ by adding 40, 35, and 25. This is equal to 100 centimeters. The perimeter of triangle πΆπ·πΈ will therefore be equal to half of this. This is equal to 50 centimeters.

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