Quadratic Challenges in Right Triangles and Distances

A right triangle has one leg that is 3 meters longer than the other leg. If the length of the shorter leg is x meters, write a quadratic equation to represent the relationship between the legs and find the lengths of both legs.

• The lengths of the legs are (2x) meters and (3x) meters.

• The lengths of the legs are (x + 3) meters and (x + 6) meters.

• The lengths of the legs are x meters and (x + 3) meters, where x is the length of the shorter leg.

• The lengths of the legs are x meters and (x - 3) meters.

The height of a triangular garden is 4 meters less than twice the base. If the area of the garden is 24 square meters, set up a quadratic equation to find the base and height of the garden.

• The base is 8 meters and the height is 10 meters.

• The base is 4 meters and the height is 6 meters.

• The base is 6 meters and the height is 8 meters.

• The base is 5 meters and the height is 7 meters.

A ladder leans against a wall, forming a right triangle with the ground. If the foot of the ladder is 2 meters away from the wall and the ladder is 10 meters long, find the height at which the ladder touches the wall using the Pythagorean theorem.

• 6√3 meters

• 8 meters

• 5 meters

• 4√6 meters

A rectangular park has a length that is 5 meters more than its width. If the diagonal of the park is 13 meters, write a quadratic equation to find the dimensions of the park.

• w^2 + 10w - 144 = 0

• 2w^2 + 10w - 144 = 0

• 2w^2 + 5w - 144 = 0

• w^2 + 15w - 144 = 0

A right triangle has one leg that measures 6 cm, and the hypotenuse is 10 cm. Calculate the length of the other leg using the Pythagorean theorem and express it as a quadratic equation.

• b^2 - 49 = 0

• b^2 - 64 = 0

• b^2 - 100 = 0

• b^2 - 36 = 0

The sum of the lengths of the two legs of a right triangle is 20 cm. If one leg is x cm long, express the length of the other leg in terms of x and set up a quadratic equation to find the lengths of both legs.

• The lengths of the legs are x cm and (20 - x) cm.

• The lengths of the legs are 10 cm and 10 cm.

• The lengths of the legs are x cm and (20 + x) cm.

• The lengths of the legs are (x + 5) cm and (15 - x) cm.

A triangular piece of land has sides measuring 8 meters, 15 meters, and x meters. Use the Pythagorean theorem to find the value of x if the triangle is a right triangle.

• 10 meters

• 14 meters

• 16 meters

• √161 or approximately 12.688 meters

A right triangle has a hypotenuse of 25 cm and one leg measuring 7 cm. Write a quadratic equation to find the length of the other leg and solve for it.

• 24 cm

• 20 cm

• 30 cm

• 15 cm

A rectangular swimming pool has a length that is twice its width. If the diagonal of the pool measures 20 meters, set up a quadratic equation to find the dimensions of the pool.

• Width = 5 meters, Length = 10 meters

• Width = 4√5 meters, Length = 8√5 meters

• Width = 3 meters, Length = 6 meters

• Width = 6 meters, Length = 12 meters

A right triangle has legs measuring x and (x + 4) cm. If the hypotenuse measures 10 cm, write a quadratic equation to find the value of x and determine the lengths of the legs.

• The lengths of the legs are 5 cm and 9 cm.

• The lengths of the legs are 6 cm and 10 cm.

• The lengths of the legs are 7 cm and 11 cm.

• The lengths of the legs are 8 cm and 12 cm.