Math Problem Statement

V pravouhlom trojuholníku ABC s pravým uhlom pri vrchole B poznáme: a = 8 cm, b = 10 cm. Vypočítať jeho obvod, obsah a vnútorné uhly. Strecha má v prieze tvar rovnoramenného trojuholníka so základňou z = 24 m. Vypočítať dĺžku bočného ramena, ak sklon strechy je 40°. Riešte trojuholník ABC, ak je dané: a = 8 cm, b = 9 cm, γ = 120°. Vypočítať výšku stožiara, ktorého pätu vidíme v hĺbkovom uhle 11° a vrchol vo výškovom uhle 29°. Stožiar je pozorovaný z miesta 10 m nad úrovňou pätý stožiara.

Solution

The uploaded image contains several math problems. Let's break them down one by one:

  1. Right Triangle ABC:

    • We know it's a right triangle, and the angle at vertex B is 90∘90^\circ. The sides are given as:
      • a=8a = 8 cm (leg)
      • b=10b = 10 cm (leg)
    • The goal is to find the perimeter, area, and internal angles of the triangle.

  2. Roof Geometry:

    • The roof has a cross-section in the shape of an isosceles triangle with the base z=24z = 24 m. The inclination angle of the roof is 40°. We need to find the length of the slanted side of the roof.

  3. Triangle ABC:

    • Given side lengths and angles: a=8a = 8 cm, b=9b = 9 cm, and γ=120∘\gamma = 120^\circ. The task is to solve for the other angles and side lengths of triangle ABC.

  4. Height of a Tower:

    • A tower's base is seen at an angle of 11° and the top at 29°. The height of the observer from the base is 10 m. We are to find the height of the tower.

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Math Problem Analysis

Mathematical Concepts

Right Triangles
Perimeter and Area Calculation
Trigonometry
Roof Geometry
Law of Cosines
Height and Angles in Elevation Problems

Formulas

Perimeter of a Right Triangle: P = a + b + c
Area of a Right Triangle: A = (1/2) * base * height
Law of Cosines: c^2 = a^2 + b^2 - 2ab * cos(γ)
Trigonometric Height Formula: height = tan(angle) * distance

Theorems

Pythagorean Theorem
Law of Cosines
Law of Sines

Suitable Grade Level

Grades 9-12

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