Mathematics 10 — Exercise Review · Q8

Full question

Use vectors and magnitudes to show the triangle is isosceles.

Step-by-step solution

1. Calculate side AB.

2. Vector from the first point to the second is endpoint minus starting point.

   $$4\times î+6\times ĵ-(î+2\times ĵ)$$

3. Subtract the matching starting coordinate from the endpoint coordinate.

   $$4-1=3$$

4. Subtract the matching starting coordinate from the endpoint coordinate.

   $$6-2=4$$

5. Write the two calculated components as a vector.

   $$3\times î+4\times ĵ$$

6. Use the horizontal and vertical components as the two perpendicular sides.

   $$\left[\begin{array}{c}3\\ 4\end{array}\right]$$

7. Magnitude is the nonnegative square root of the sum of component squares.

   $$\sqrt{3^2+4^2}$$

8. Square the horizontal component, including its sign.

   $$3^2=9$$

9. Square the vertical component, including its sign.

   $$4^2=16$$

10. Add these two squares.

    $$9+16=25$$

11. Take and simplify the positive square root.

    $$\sqrt{25}=5$$

12. Calculate side BC.

13. Vector from the first point to the second is endpoint minus starting point.

    $$7\times î+2\times ĵ-(4\times î+6\times ĵ)$$

14. Subtract the matching starting coordinate from the endpoint coordinate.

    $$7-4=3$$

15. Subtract the matching starting coordinate from the endpoint coordinate.

    $$2-6=-4$$

16. Write the two calculated components as a vector.

    $$3\times î-4\times ĵ$$

17. Use the horizontal and vertical components as the two perpendicular sides.

    $$\left[\begin{array}{c}3\\ -4\end{array}\right]$$

18. Magnitude is the nonnegative square root of the sum of component squares.

    $$\sqrt{3^2+{\left(-4\right)}^2}$$

19. Square the horizontal component, including its sign.

    $$3^2=9$$

20. Square the vertical component, including its sign.

    $${\left(-4\right)}^2=16$$

21. Add these two squares.

    $$9+16=25$$

22. Take and simplify the positive square root.

    $$\sqrt{25}=5$$

23. Calculate side CA.

24. Vector from the first point to the second is endpoint minus starting point.

    $$î+2\times ĵ-(7\times î+2\times ĵ)$$

25. Subtract the matching starting coordinate from the endpoint coordinate.

    $$1-7=-6$$

26. Subtract the matching starting coordinate from the endpoint coordinate.

    $$2-2=0$$

27. Write the two calculated components as a vector.

    $$-6\times î+0\times ĵ$$

28. Use the horizontal and vertical components as the two perpendicular sides.

    $$\left[\begin{array}{c}-6\\ 0\end{array}\right]$$

29. Magnitude is the nonnegative square root of the sum of component squares.

    $$\sqrt{{\left(-6\right)}^2+0^2}$$

30. Square the horizontal component, including its sign.

    $${\left(-6\right)}^2=36$$

31. Square the vertical component, including its sign.

    $$0^2=0$$

32. Add these two squares.

    $$36+0=36$$

33. Take and simplify the positive square root.

    $$\sqrt{36}=6$$

34. Two of these calculated side lengths are equal, so the triangle is isosceles.

    $$(5,5,6)$$

Answer

Isosceles triangle.