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Chapter 14: Problem 13

Construct an isosceles triangle, given the vertex angle and the altitude to the base.

Short Answer

Expert verified
In this construction, one starts by drawing a base line. From a point on the baseline as the vertex, the perpendicular altitude is drawn. The two equal sides of the triangle are then drawn from the vertex, forming the prescribed angle that the corresponding lines make with the baseline. The constructed figure is then checked to be an isosceles triangle with the given vertex angle and altitude.

Step by step solution

01

Draw Base Line

Begin by drawing a straight horizontal line, which will act as the base of the triangle. The length of this line is not given and doesn't matter as it won't change the vertex angle or altitude.
02

Draw Altitude

Mark a point (let's call it 'A') anywhere on the line. This point will serve as the vertex of the triangle. From point 'A', raise a perpendicular line upwards as per the given length of the altitude. This line is called AD.
03

Draw the Sides of the Triangle

Using a protractor, measure the given vertex angle at point 'A'. Draw two lines from 'A' to the baseline to create this angle. These two lines, which should be of equal length, form the two equal sides of the isosceles triangle. Let's name these points as 'B' and 'C', forming triangle ABC.
04

Check the Triangle

Ensure that triangle ABC is indeed an isosceles triangle, i.e., AB = AC and angle BAC is as prescribed. Simultaneously, verify the length of the altitude AD is correct. If these conditions hold true, then you've correctly constructed the triangle.

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Key Concepts

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

Understanding the Vertex Angle
The vertex angle in an isosceles triangle is one of the key elements that define its structure. It is the angle formed between the two equal sides at the apex of the triangle. In the construction of an isosceles triangle, knowing the vertex angle is essential because it determines how sharp or wide the triangle's peak will be.
To visualize this, imagine at point 'A' of our previously described triangle ABC, the lines AB and AC meet to form this vertex angle. This angle influences the overall shape and should be measured accurately to ensure the triangle maintains its isosceles properties.
  • The vertex angle is situated opposite the base of the triangle.
  • In an equilateral triangle, which is a special case of isosceles, the vertex angle is 60 degrees.
Defining the Altitude
The altitude in a triangle is a line segment from a vertex perpendicular to the opposite side or its extension. In the context of constructing an isosceles triangle, the altitude is an invaluable line as it helps ensure symmetry and correctness in the triangle's form.
For our isosceles triangle ABC, the altitude is the line segment AD, drawn from the vertex point 'A' straight down to the base line BC. It should meet the base at a right angle, ensuring that the triangle is evenly divided.
  • Altitudes can sometimes be outside the triangle if the base is extended, especially in obtuse triangles.
  • It acts as an axis of symmetry for the isosceles triangle.
  • In isosceles and equilateral triangles, the altitude also bisects the vertex angle and the base.
Using a Protractor Effectively
A protractor is a simple yet important tool used in geometry for measuring angles. When constructing an isosceles triangle, using a protractor is crucial for accurately measuring the vertex angle.
To use the protractor in constructing triangle ABC, you would place the midpoint of your protractor at point 'A', lining it up straight with the base line BC. Measuring the given vertex angle precisely ensures that lines AB and AC are drawn accurately, keeping the triangle balanced.
  • Ensure the protractor is aligned correctly before taking measurements.
  • Always check your measured angles for precision as this affects the overall shape and balance of the triangle.
  • Use a protractor that displays angles in degrees from 0 to 180 for best results.
Exploring Isosceles Triangle Properties
Isosceles triangles are fascinating because of their distinct properties and symmetry. A major feature of an isosceles triangle is two sides of equal length, which influence its geometry and characteristics.
Key properties of isosceles triangles include:
  • Equal Sides: The lengths of two sides are consistent and equal, which helps to maintain balance.
  • Equal Angles: The angles opposite the equal sides, called base angles, are also equal. Understanding this helps in geometric problem-solving.
  • Symmetrical Alignment: The altitude drawn from the apex to the base bisects the isosceles triangle into two congruent right triangles, revealing its symmetrical nature.
These properties make isosceles triangles a popular study subject in geometry, aiding in the understanding of congruence and proofs.

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Most popular questions from this chapter

Recall that the coordinates of the midpoint of a side of a triangle are the averages of the coordinates of the endpoints. As an extension of this idea, it can be shown that the coordinates of the centroid of a triangle are the averages of the coordinates of the three vertices of the triangle. Given: \(\triangle \mathrm{ABC},\) with \(\mathrm{A}=(-2,8), \mathrm{B}=(-6,-2),\) and \(\mathrm{C}=(12,6)\) Find: a The coordinates of the centroid of \(\triangle \mathrm{ABC}\) b The coordinates of the centroid of the triangle formed by joining the midpoints of the sides of \(\triangle \mathrm{ABC}\)

Find the locus of points equidistant from two concentric circles and on a diameter of the larger circle.

Sketch a triangle and its medians. As you know, the centroid of the triangle is one of the trisection points of each median. Now form another triangle by joining the other trisection points of the medians. a Find the ratio of the area of this triangle to the area of the original triangle. b What is the relationship of this triangle to the triangle formed by joining the midpoints of the sides of the original triangle.

Construct a triangle, given the three medians.

Construct an isosceles trapezoid, given the bases and the altitude.
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