The right triangle altitude theorem - practice problems
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):
h2=c1c2
Or: The height in a right triangle is the geometric mean of two sections of the hypotenuse.
h=c1⋅c2
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.
a2=c⋅c1
b2=c⋅c2
Or: The hypotenuse of a right triangle is the geometric diameter of the hypotenuse and the adjacent section of the hypotenuse.
a=c⋅c1
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: 64
- An isosceles triangle
An altitude is drawn from the vertex of an isosceles triangle, forming a right angle and two congruent triangles. As a result, the altitude cuts the base into two equal segments. The length of the altitude is 18 inches, and the length of the base is 15 in - Euclid2
The ABC right triangle with a right angle at C is side a=29 and height v=17. Calculate the perimeter of the triangle. - Isosceles IV
In an isosceles triangle ABC is |AC| = |BC| = 13 and |AB| = 10. Calculate the radius of the inscribed (r) and described (R) circle. - Height of right RT
The right triangle ABC has a hypotenuse c 9 cm long and a part of the hypotenuse cb = 3 cm. How long is the height of this right triangle?
- Leg and height
Solve right triangle with height v = 9.6 m and shorter cathetus b = 17.3 m. - Hypotenuse and height
In a right triangle is length of the hypotenuse c = 56 cm and height hc = 4 cm. Determine the length of both triangle legs. - Euclid 5
Calculate the length of remain sides of a right triangle ABC if a = 7 cm and height vc = 5 cm. - Euklid4
The legs of a right triangle have dimensions 241 m and 34 m. Calculate the length of the hypotenuse and the height of this right triangle. - Triangle ABC
Right triangle ABC with right angle at the C, |BC|=19, |AB|=32. Calculate the height of the triangle hAB to the side AB.
- Euclid3
Calculate the height and sides of the right triangle if one leg is a = 81 cm and the section of hypotenuse adjacent to the second leg cb = 39 cm. - Circle in rhombus
In the rhombus is an inscribed circle. Contact points of touch divide the sides into parts of length 14 mm and 9 mm. Calculate the circle area. - Proof PT
Can you easily prove Pythagoras' theorem using Euclidean theorems? If so, do it. - Area of RT
The right triangle has orthogonal projections of legs to the hypotenuse lengths 15 cm and 9 cm. Determine the area of this triangle. - Euclid1
The right triangle has hypotenuse c = 27 cm. How large sections cuts height hc=3 cm on the hypotenuse c?
- RT triangle and height
Calculate the remaining sides of the right triangle if we know side b = 4 cm long and height to side c h = 2.4 cm. - Spruce height
How tall was spruce that was cut at an altitude of 8m above the ground and the top landed at a distance of 15m from the heel of the tree? - Without Euclid laws
Right triangle ABC with a right angle at the C has a=14 and hypotenuse c=26. Calculate the height h of this triangle without the use of Euclidean laws. - Area of RT
Calculate the right triangle area that hypotenuse has length 14, and one hypotenuse segment has length 5. - RT - hypotenuse and altitude
The right triangle BTG has hypotenuse g=117 m, and the altitude to g is 54 m. How long are hypotenuse segments?
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