Triangle Properties and Theorems

In this video, the problem of finding the area of a right triangle with an inscribed circle is tackled. The video begins with an introduction to the problem and the setup of the triangle. The two tangent theorem is explained and applied to determine congruent line segments. The dimensions of the triangle are calculated using given lengths and theorems. A quadratic equation is derived and solved to find the radius of the inscribed circle. Finally, the area of the triangle is calculated using the derived values, resulting in an area of 135 square units.

What is the length of line segment CD in the given problem?

• 24 units

• 15 units

• 9 units

• 12 units

Which theorem states that two tangent segments from the same external point to a circle are congruent?

• Circle Theorem

• Congruent Segments Theorem

• Two Tangent Theorem

• Pythagorean Theorem

If AD is 15 units, what is the length of AE according to the Two Tangent Theorem?

• 15 units

• 12 units

• 24 units

• 9 units

What is the total length of the longest leg AC in the triangle?

• 15 units

• 18 units

• 24 units

• 9 units

What formula is used to calculate the dimensions of the right triangle?

• Area formula

• Pythagorean theorem

• Quadratic formula

• Two Tangent Theorem

What is the quadratic equation derived to find the radius of the circle?

• r² + 24r - 135 = 0

• r² + 48r - 270 = 0

• r² + 30r - 225 = 0

• r² + 18r - 81 = 0

What is the base of the triangle ABC after substituting the radius?

• 15 + 3 * sqrt(31)

• 3 * sqrt(31) - 3

• 9 + 3 * sqrt(31)

• 3 * sqrt(31) + 3

What is the height of the triangle ABC after substituting the radius?

• 3 * sqrt(31) + 3

• 9 + 3 * sqrt(31)

• 3 * sqrt(31) - 3

• 15 + 3 * sqrt(31)

What is the formula used to calculate the area of the triangle?

• Base * Height

• 1/2 * Base * Height

• Base + Height

• Base - Height

What is the final area of the triangle ABC?

• 180 square units

• 135 square units

• 270 square units

• 150 square units