sk|cz|

# Right Triangle

1. Define a right triangle.

Solution:

a) Pythagorean theorem:

a2 + b2 = c2

b) Goniometric functions:

c) Perimeter:

O = a + b + c

d) Area:

c) Euclid theorems:

a2 = c.ca
b2 = c.cb
v2 = ca.cb

2. The catheti of a right triangle ΔABC are a =3 cm, b = 4 cm.

Determine:
a) Size of the hypotenuse c
b) Height perpendicular to the hypotenuse, vc
c) Area S
d) Acute angles α and β
Solution:

For the triangle ΔABC stands: c = 5cm, vc = 2,4cm, S = 6cm2, α = 36,87°, β = 53,16°.

3. Let a, b be a right triangles‘ catheti, c a right triangles‘ hypotenuse, α the angle opposite to a.

Fill the table! (Lengths in cm)

Solution:

4.The sides of a right triangle make an arithmetic progression. Determine:

a) perimeter
b) area
of the triangle. Size of the longer cathetus equals 16 cm.
Solution:

a = 16 - x
b = 16
c = 16 + x

(16 – x)2 + 162 = (16 + x)2
256 – 32x + x2 + 256 = 256 + 32x + x2
64x = 256
x = 4

a) Perimeter

a = 12
b = 16
c = 20

O = a + b + c
O = 12 +16+20
O = 48 cm

b) Area

S = 0,5.a.b
S = 0,5.12.16
S = 96 cm2

The perimeter of the triangle is O = 48 cm, the area of the triangle is S = 96 cm2.

5.Sum of the lengths of the catheti of a right triangle is 30 cm. The area of the triangle is 110,5 cm2. Determine the perimeter of the triangle.

Solution:

The perimeter of the triangle is 51,4 cm.

6. The leg of an isosceles triangle is 1 cm longer than it‘s base. The altitude is 2 cm shorter than the leg. Determine the size of the leg of the triangle.

Solution:

z = 16 cm, r = 17 cm, v = 15cm

7. A triangle‘s sides lengths equal a = 4p2- 1, b = 4p, c = 4p2+1. Prove that it‘s a right triangle and that it‘s a Pythagorean triangle. Write out four Pythagorean triangles.

Solution:

If a Pythagorean theorem stands for a triangle, it‘s a right triangle.

The triangle is a right triangle.

Pythagorean triangles:

Proof:
352 + 122 = 372
1225 + 144 = 1369
1369 = 1369

8. Aké stúpanie má cesta, ak na dopravnej značke, ktorá o tom informuje, je napísané 6,7 %? Auto prešlo 2,3 km po tejto ceste.

Aký výškový rozdiel auto prekonalo?
Riešenie:

Uhol stúpania cesty je 3,83°.
Auto prekonalo výškový rozdiel asi 154 m.

9. Vypočítajte obsah rovnoramenného pravouhlého trojuholníka, ktorého obvod je 20cm.

Riešenie:

Obsah trojuholníka je 17,15 cm2.

10. Pre odvesny pravouhlého trojuholníka platí a:b = 2:3. Prepona má dĺžku 10 cm.

Vypočítajte obvod a obsah tohto trojuholníka.
Riešenie:

Obvod trojuholníka ΔABC je 23,86 cm a obsah 23,045 cm2.