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Chapter 14: Problem 17

Construct a right triangle, given the hypotenuse and the altitude to the hypotenuse.

Short Answer

Expert verified
To construct a right triangle given the hypotenuse and altitude to the hypotenuse, draw a line of the length of the hypotenuse, construct a line perpendicular to this of the length of the altitude, and connect this endpoint with the other end of the hypotenuse.

Step by step solution

01

Draw the Hypotenuse

Begin by drawing a straight line of the length of the given hypotenuse.
02

Construct the Perpendicular

At one endpoint of the hypotenuse, construct a perpendicular line to the hypotenuse with length equal to the given altitude.
03

Connect the Endpoints

From the endpoint of the constructed altitude, draw a line segment to the other endpoint of the hypotenuse.
04

Confirm the Triangle

The final figure is a right triangle, with the drawn hypotenuse, constructed altitude as one leg of the right triangle, and the line segment as the second leg.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypotenuse
In the realm of right triangles, the hypotenuse is a key player. It's the longest side of a right triangle and can be found opposite the right angle. In construction, it serves as a critical foundation. To begin a right triangle construction, one starts by drawing this longest side, the hypotenuse.
  • The length of the hypotenuse is often provided in exercises or tasks.
  • Begin your construction with precision by drawing the hypotenuse as a straight line.
  • The other two sides of a right triangle (called the legs) are defined in relation to the hypotenuse.
Understanding the hypotenuse is essential to successfully build and analyze geometric figures. Once you've acknowledged its role, it's easier to wrap your head around how it interconnects with the other parts of the triangle.
Altitude
The term altitude might remind you of height, and that's precisely its role in geometry as well. The altitude of a right triangle is a perpendicular line from the hypotenuse to the opposite vertex. It's like a height that drops straight down.
  • This perpendicular line is often referred to as a height of the triangle.
  • In a right triangle, one leg may serve as the altitude to the hypotenuse.
  • Construction of this altitude begins at an endpoint of the hypotenuse and extends perpendicularly.
Knowing how to construct this line is crucial, especially when working out problems involving heights or areas of a triangle. It becomes the basis for calculating geometric properties effectively.
Perpendicular Line
A perpendicular line forms an angle of 90 degrees with another line. In the context of right triangles, and geometry in general, this concept is a fundamental tool.
  • In geometry exercises, you are often asked to draw a perpendicular line to find other lengths or angles.
  • When constructing a right triangle, a perpendicular line from the hypotenuse is pivotal to define the right angle.
  • Tools such as set squares or compass-and-straightedge constructions are often used to accurately draw perpendicular lines.
Mastering the art of drawing perpendicular lines not only helps in creating perfect right triangles but also enhances your overall understanding of geometric relationships. Recognizing and utilizing these lines aids in simplifying complex geometric problems.

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Most popular questions from this chapter

Every triangle has a circumcenter, an orthocenter, a centroid, and an incenter. Which of the four points will always lie in the interior of the triangle?

Construct the centroid of a given triangle.

Construct the three altitudes of an acute \(\triangle \mathrm{ABC}\).

Construct an isosceles right triangle, given the median to the hypotenuse.

Construct a rectangle, given the base and a diagonal.
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