# The right triangle altitude theorem - practice problems - page 3 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: 63

- Triangle KLM

In the rectangular triangle KLM, where is hypotenuse m (sketch it!). Find the length of the leg k and the height of triangle h if the hypotenuse's segments are known MK = 5cm and ml = 15 cm. - Hypotenuse, euclid

In a right-angled triangle, the hypotenuse has a length of 24 cm. The heel of the height on the hypotenuse divides it into two parts in a ratio of 2:4. What size in cm is the height at the hypotenuse? Calculate the perimeter of this right triangle in cent - Altitude angles

Cities A, B, and C lie in one elevation plane. C is 50 km east of B, and B is north of A. C is deviated by 50° from A. The plane flies around places A, B, and C at the same altitude. When the aircraft is flying around B, its altitude angle to A is 12°. Fi - Free space in the garden

The grandfather's free space in the garden was in the shape of a rectangular triangle of 5 meters and 12 meters in length. He decided to divide it into two parts and the height of the hypotenuse. The smaller part creates a rock garden, for the larger sows

- Squares above sides

Two squares are constructed on two sides of the ABC triangle. The square area above the BC side is 25 cm². The height vc to the side AB is 3 cm long. The heel P of height vc divides the AB side in a 2: 1 ratio. The AC side is longer than the BC side. Calc - Circumscribed 6568

In a right triangle ABC with a right angle at the vertex C, it is given: a = 17cm, Vc = 8 cm. Calculate the length of the sides b, c, its area S, the perimeter o, the length of the radii of the circles of the triangle circumscribed by R and inscribed r an - Isosceles triangle 9

There is an isosceles triangle ABC where AB= AC. The perimeter is 64cm, and the altitude is 24cm. Find the area of the isosceles triangle. - Goat and circles

What is the radius of a circle centered on the other circle, and is the intersection of the two circles equal to half the area of the first circle? This task is the mathematical expression of the role of agriculture. The farmer has circular land on which - Observatory 71934

The aircraft flying towards the observatory was aimed at a distance of 5300 m at an elevation angle of 28º and after 9 seconds at a distance of 2400 m at an elevation angle of 50º. Calculate the distance the plane has flown in this time interval, its spee

- MIT 1869

You know the length of hypotenuse parts 9 and 16, at which the hypotenuse of a right triangle is divided by a height. The task is to find the lengths of the sides of the triangle and the length of line x. This assignment was part of the Massachusetts Inst - Paratrooper

After the parachute is opened, the paratrooper drops to the ground at a constant speed of 2 m/s, with the sidewinding at a steady speed of 1.5 m/s. Find: a) the magnitude of its resulting velocity concerning the ground, b) the distance of his land from a - Euclid theorems

Calculate the sides of a right triangle if leg a = 6 cm and a section of the hypotenuse, which is located adjacent to the second leg b, is 5cm. - The bomber

An aircraft flying at an altitude of 1260 m. From what distance in front of the target must a parachute load be dropped from an airplane? The load slopes at a speed of 5.6 m/s and moves in the direction of movement at 12 m/s. What is the direct distance o - An observer

An observer standing west of the tower sees its top at an altitude angle of 45 degrees. After moving 50 meters to the south, he sees its top at an altitude angle of 30 degrees. How tall is the tower?

- Circles

In the circle with a radius, 7.5 cm is constructed of two parallel chords whose lengths are 9 cm and 12 cm. Calculate the distance of these chords (if there are two possible solutions, write both). - Square equal rhombus

Construct a square that has the same area as a rhombus ABCD if |AB| = 5cm, |AD| = 4cm and angle |DAB| = 30°. - Calculate 16223

The following elements are known in the right triangle ABC: a = 10 cm, height to side c h = 9.23 cm. Calculate o, R (radius of the inscribed circle), r (radius of the inscribed circle). - Touch circle

Point A has a distance (A, k) = 10 cm from a circle k with radius r = 4 cm and center S. Calculate: a) the distance of point A from the point of contact T if the tangent to the circle is drawn from point A b) the distance of the contact point T from the l - Hypotenuse - RT

A triangle has a hypotenuse of 55 and an altitude to the hypotenuse of 33. What is the area of the triangle?

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