How to Construct a Right-Angled Triangle with Given Hypotenuse and Included Angle

Constructing a right-angled triangle with a specified hypotenuse and an included angle can be a precise process. Here, we will guide you step-by-step on how to create a right-angled triangle with hypotenuse AC 10 cm, side BC 6 cm, and the included angle ∠ABC 50°. We will also address the inconsistency that arises when attempting to build the triangle with the given measurements.

Materials Needed

Compass Ruler Protractor Pencil Paper

Steps to Construct the Triangle

Draw the Hypotenuse AC: Use the ruler to draw a line segment AC that is 10 cm long. Label the endpoints as A and C. Construct Angle ABC: At point B (which will be one endpoint of the right angle), use the protractor to measure an angle of 50° from line AC. Mark this angle with a light pencil line extending from point B. Determine Point B's Position: Since triangle ABC is a right triangle with ∠ABC 50°, you will need to find point B such that BC 6 cm. From point B on the angle line, use the ruler to measure 6 cm along the line towards the direction of the angle you just drew. Mark this point as B. Complete Triangle ABC: Now connect point B to points A and C using straight lines to form triangle ABC. Check the Right Angle: Ensure that ∠ABC is indeed a right angle by checking that it measures 90° using the protractor. Alternatively, you can use the Pythagorean theorem to confirm that the lengths are correct: AB2 BC2 AC2. Calculate AB using the cosine of the angle: AB AC · cos50° ≈ 10 · 0.643 ≈ 6.43 cm. Confirm: 6.432 62 ≈ 41.38 36 77.38 and 102 100. This shows the triangle is constructed correctly but you may need to adjust for exact lengths based on your measurements.

Addressing the Inconsistency

The method described above does not yield a right-angled triangle when using the specified measurements. This is because the triangle with the given dimensions (hypotenuse AC 10 cm, side BC 6 cm, and included angle ∠ABC 50°) does not satisfy the properties of a right-angled triangle.

A right-angled triangle with the given hypotenuse AC 10 cm and angle ∠ABC 50° would have a vertical side AB of approximately 7.66 cm (AB 10 sin50° ≈ 7.66 cm) and a base BC of approximately 6.43 cm (BC 10 cos50° ≈ 6.43 cm). Therefore, if we were to construct a right-angled triangle ABC with hypotenuse AC 10 cm and base BC 6 cm, the vertical side AB would need to be 8 cm (AB 8 cm), and the included angle would be arcsin(0.8) ≈ 53.1°.

If a right triangle with the specified base BC 6 cm and vertical side AB 8 cm is required, the hypotenuse AC would be calculated as follows:

AC √(AB2 BC2) √(82 62) √(64 36) √100 10 cm.

Therefore, to construct a right-angled triangle ABC with the specified dimensions and angles, the measurements need to be adjusted.

Conclusion

In summary, when attempting to construct a right-angled triangle with a given hypotenuse and included angle, it is essential to ensure that the triangle adheres to the properties of a right-angled triangle and that the measurements are consistent with the given angles. Any discrepancies may indicate the need to adjust the measurements for accurate construction.

Frequently Asked Questions (FAQ)

Q: Can a right-angled triangle be constructed with the given measurements? A: No, the provided measurements do not yield a right-angled triangle. The heights and bases of the triangle need adjustment for a right-angled triangle. Q: How can I check if a triangle is right-angled using a protractor? A: Measure each angle; a right angle should measure 90°. Q: What is the Pythagorean theorem, and how is it used in constructing a right-angled triangle? A: The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This is used to verify the correctness of the triangle's dimensions.