How to Construct an Isosceles Triangle with Given Base and Vertical Angle

Construction of an isosceles triangle, where you know the base length and the vertical angle, can be a fundamental skill in geometry. This process involves precise measurement and use of basic drafting tools. Below, we guide you through the step-by-step procedure, including the necessary materials and tools you will need.

Materials Needed

Compass Straightedge ruler without markings Pencil (Optional) Protractor

Step-by-Step Instructions

Step 1: Draw the Base

Use the straightedge to draw a horizontal line segment of the desired base length. Label the endpoints as A and B.

Step 2: Find the Midpoint of the Base

Locate the midpoint M of the line segment AB. You can do this by measuring the length of AB and marking the point that is half that distance from either A or B.

Step 3: Construct the Vertical Angle

Use a protractor to measure the vertical angle θ (theta).

From point M, draw a line perpendicular to AB. You can do this by measuring 90° from AB using your protractor or by folding the paper if you are using physical construction.

Step 4: Determine the Apex Point

Using the compass, measure the distance from M to where you want the apex of the triangle to be along the vertical line. This distance will depend on the angle θ and the desired height of the triangle.

Mark that point as C where the vertical line intersects the desired height.

Step 5: Connect the Points

Use the straightedge to draw line segments AC and BC.

As a result, you will have constructed an isosceles triangle ABC with base AB where AC BC and the angle ∠ACB is the specified vertical angle θ.

Additional Tips

If you need to ensure that the sides AC and BC are equal, you can measure the distance from M to C and use that distance to ensure both sides are equal when drawing the lines. Additionally, the vertical height can also be calculated using trigonometric functions if needed depending on your angle and base length.

Example Construction

Let's say you want to draw an isosceles triangle with a base of 10 cm and a vertical angle of 80°. The base angles in the isosceles triangle will be (180° - 80°)/2 each, which is 100°/2 50°.

Step 1: Draw the Base

Draw a line 10 cm long and mark the ends as B and C.

Step 2: Assign Angles at B and C

With a protractor, make angles of 50° at the two ends B and C.

Step 3: Locate the Vertex A

Wherever the two rays intersect will be the vertex A, and angle BAC will be 80°. Now you have the isosceles triangle of base 10 cm and vertical angle of 80°.