Question Video: Using the Sine Rule to Find the Length of A Side of A Triangle in A Composite Figure | Nagwa Question Video: Using the Sine Rule to Find the Length of A Side of A Triangle in A Composite Figure | Nagwa

Question Video: Using the Sine Rule to Find the Length of A Side of A Triangle in A Composite Figure Mathematics • Second Year of Secondary School

Join Nagwa Classes

Attend live General Mathematics sessions on Nagwa Classes to learn more about this topic from an expert teacher!

In the given figure, π΅πΆπ·π‘Œ is a rectangle and 𝐡 is a point on the straight line 𝐴𝐢. 𝐡𝐢 = 405 m, π‘šβˆ π·π΄πΆ = 21Β°, and π‘šβˆ π‘Œπ΄πΆ = 59Β°. Find the length of 𝐷𝐢 giving the answer to the nearest meter.

04:10

Video Transcript

In the given figure, π΅πΆπ·π‘Œ is a rectangle and 𝐡 is a point on the straight line 𝐴𝐢. 𝐡𝐢 is equal to 405 meters, the measure of angle 𝐷𝐴𝐢 is 21 degrees, and the measure of angle π‘Œπ΄πΆ is 59 degrees. Find the length of line segment 𝐷𝐢, giving the answer to the nearest meter.

We will begin by adding the information from the question onto our diagram. As π΅πΆπ·π‘Œ is a rectangle, all the angles inside it will be equal to 90 degrees. We are told that 𝐡 is a point on the straight line 𝐴𝐢. Therefore, the angle π‘Œπ΅π΄ is also 90 degrees. The measure of angle 𝐷𝐴𝐢 is 21 degrees. And since the measure of angle π‘Œπ΄πΆ is 59 degrees, then the measure of angle π‘Œπ΄π· is 59 degrees minus 21 degrees. This is equal to 38 degrees. Finally, we are told that 𝐡𝐢 has length 405 meters, which means that π‘Œπ· is also equal to 405 meters.

Since angles in a triangle sum to 180 degrees, the measure of angle π΄π‘Œπ΅ is 31 degrees. We are asked to find the length of line segment 𝐷𝐢, which we have labeled π‘₯. There are many ways to do this. One way would be to split our shape into two triangles, π΄π‘Œπ· and 𝐴𝐢𝐷. We know that one way to calculate missing lengths in triangles is using the sine rule or law of sines. This states that π‘Ž over sin 𝐴 is equal to 𝑏 over sin 𝐡, which is equal to 𝑐 over sin 𝐢. Applying this rule to triangle π΄π‘Œπ·, we have side length 𝐴𝐷 divided by sin of 121 degrees is equal to 405 divided by sin of 38 degrees. Multiplying through by the sin of 121 degrees, we can calculate the length of side 𝐴𝐷. This is equal to 563.8695 and so on. As this is not the final answer, we will not round. But we know that the length of 𝐴𝐷 is 563.8695 and so on meters.

We can now consider triangle 𝐴𝐷𝐢. We could use the sine rule once again as we know the measure of two angles together with the length of one side. However, since this is a right triangle, we can simply use the sine ratio, which states that the sin of angle πœƒ is equal to the opposite over the hypotenuse. The sin of 21 degrees is equal to π‘₯ over 563.8695 and so on. Multiplying through by the denominator, we have π‘₯ is equal to 202.0727 and so on. We are asked to give our answer to the nearest meter. Since the first digit after the decimal point is zero, we round down. The length of 𝐷𝐢 correct to the nearest meter is 202 meters.

Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

  • Interactive Sessions
  • Chat & Messaging
  • Realistic Exam Questions

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy