# The right triangle altitude theorem - practice problems - page 2 of 4

Euclid was a Greek mathematician and philosopher. He left us with two important but simple theorems that apply in a right triangle.**Euclid's first theorem (about height)**: The area of the square constructed above the height of the right triangle (h) is equal to the area of the rectangle constructed from both sections of the hypotenuse (c1 and c2):

$h_{2}=c_{1}c_{2}$

Or: The height in a right triangle is the geometric mean of two sections of the hypotenuse.

$h=c_{1}⋅c_{2} $

**Euclid's second theorem - about the hypotenuse**: The area of the square constructed above the hypotenuse of a right-angled triangle is equal to the area of the rectangle constructed from the hypotenuse and the segment of the hypotenuse adjacent to this hypotenuse.

$a_{2}=c⋅c_{1}$

$b_{2}=c⋅c_{2}$

Or: The hypotenuse of a right triangle is the geometric diameter of the hypotenuse and the adjacent section of the hypotenuse.

$a=c⋅c_{1} $

It is usually taught in high school. The Pythagorean theorem can be easily proved using Euclid's theorems.

Direction: Solve each problem carefully and show your solution in each item.

#### Number of problems found: 66

- Euclid 5

Calculate the length of remain sides of a right triangle ABC if a = 7 cm and height v_{c}= 5 cm. - RT sides

Find the sides of a rectangular triangle if legs a + b = 17cm and the radius of the written circle ρ = 2cm. - Right 24

The right isosceles triangle has an altitude x drawn from the right angle to the hypotenuse dividing it into two unequal segments. The length of one segment is 5 cm. What is the area of the triangle? Thank you. - Euclid1

The right triangle ABC has hypotenuse c = 20 cm. How large sections cuts height h_{c}=9 cm on the hypotenuse c?

- Perpendicular 32733

Calculate the right triangle ABC, the perpendicular b = 43.5 cm of the hypotenuse c = 72.9 cm. Calculate: A hypotenuse segment cb, side a, a hypotenuse segment ca, and a height of triangle v - Rectangle

In rectangle ABCD with sides, |AB|=19, |AD|=19 is from point A guided perpendicular to the diagonal BD, which intersects at point P. Determine the ratio r = (|PB|)/(|DP|). - Intersection 83575

Given a circle with a radius r = 4 cm and a point A for which |AS| applies = 10 cm. Calculate the distance of point A from the intersection of the points of contact of the tangents drawn from point A to the circle. - Quadrilateral 78874

Given is a quadrilateral ABCD inscribed in a circle, with the diagonal AC being the circle's diameter. The distance between point B and the diameter is 15 cm, and between point D and the diameter is 18 cm. Calculate the radius of the circle and the perime - Perpendicular 5667

The perpendicular projections hung on the diaphragm are 3.1 cm and 6.3 cm long in a right triangle. Calculate the perimeter of this triangle. The result is rounded to the nearest hundredth of an inch.

- Right triangle - ratio

The lengths of the legs of the right triangle ABC are in ratio b = 2:3. The hypotenuse is 10 cm long. Calculate the lengths of the legs of that triangle. - Right isosceles triangle

The right isosceles triangle has an altitude x drawn from the right angle to the hypotenuse dividing it into two equal segments. The length of one segment is 5 cm. What is the area of the triangle? - Rhombus and inscribed circle

It is given a rhombus with side a = 6 cm and the radius of the inscribed circle r = 2 cm. Calculate the length of its two diagonals. - Right-angled 82471

The lengths a = 7.2 cm and b = 10.4 cm are dropped in the right-angled triangle ABC. Do the math a) lengths of the sections of the hypotenuse b) height on the hypotenuse c - Tangents

To circle with a radius of 41 cm from the point R guided two tangents. The distance of both points of contact is 16 cm. Calculate the distance from point R and circle center.

- Right-angled 81989

Using Euclid's Theorems and Pythagoras' Theorem, complete the following parameters describing a right-angled triangle ABC with a right angle at vertex C if we know b=10, cb=8 - Triangle ABC

In a triangle ABC with the side BC of length 2 cm. Point K is the middle point of AB. Points L and M split the AC side into three equal lines. KLM is an isosceles triangle with a right angle at point K. Determine the lengths of the sides AB, AC triangle A - Right-angled 81019

In the right-angled triangle ABC (AB is the hypotenuse), a : b = 24 : 7, and the height to the side c = 12.6 cm applies. Calculate the lengths of the sides of triangle ABC. - Rhombus

It is given a rhombus of side length a = 20 cm. Touchpoints of inscribed circle divided his sides into sections a_{1}= 13 cm and a_{2}= 7 cm. Calculate the radius r of the circle and the length of the diagonals of the rhombus. - Hypotenuse 72524

We know the height of the hypotenuse h = 4cm and the hypotenuse c = 19cm in a right triangle. How to calculate the segments of legs - sections on the hypotenuse c1, c2

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