Village B is 7km due south of village A and village C is 9km due south B. Village D is 8km from A and is due east of B.
Find the distance of D from (a)B (b)C
Show workings
(a) Well, if village D is due east of B, that means D is to the right of B. Since the distance from A to B is 7km, and D is 8km from A, we can imagine a right-angled triangle where the hypotenuse is the distance from A to D, the base is the distance from A to B, and the height is the distance from B to D. Now, using some Pythagorean magic, we can calculate the distance from B to D. Is everyone ready? Drum roll, please... The distance from B to D is approximately 4.9 kilometers! A round of applause for math!
(b) Now, village C is 9km due south of B. Since we already know the distance from A to B is 7km, we can imagine another right-angled triangle (math loves triangles) with the distance from B to C as the base and the distance from A to B as the hypotenuse. Using the Pythagorean theorem again, we find that the distance from C to D is approximately 7.5 kilometers. I must say, geometry can really go the distance when it comes to finding distances!
To find the distance of village D from village B and C, we can use the Pythagorean theorem.
(a) Distance of D from B:
Considering that D is due east of B, we can form a right triangle connecting A, B, and D. The distance from A to B is 7 km, and the distance from A to D is 8 km.
Using the Pythagorean theorem, we can find the distance of D from B:
B-D distance = -√((AB distance)^2 - (AD distance)^2)
B-D distance = -√((7 km)^2 - (8 km)^2)
B-D distance = -√(49 km^2 - 64 km^2)
B-D distance = -√(-15 km^2)
B-D distance = √15 km (approximately 3.87 km)
Therefore, the distance of D from B is approximately 3.87 km.
(b) Distance of D from C:
Considering that C is due south of B, we can form a right triangle connecting B, C, and D. The distance from B to C is 9 km, and we already know the distance from B to D, which is 3.87 km.
Using the Pythagorean theorem, we can find the distance of D from C:
C-D distance = √((BC distance)^2 - (BD distance)^2)
C-D distance = √((9 km)^2 - (3.87 km)^2)
C-D distance = √(81 km^2 - 15 km^2)
C-D distance = √66 km (approximately 8.12 km)
Therefore, the distance of D from C is approximately 8.12 km.
To find the distance of point D from point B, we need to use the Pythagorean theorem, as we have a right-angled triangle formed by points A, B, and D.
a) Distance of D from B:
We know that the distance from A to B is 7km, and the distance from A to D is 8km. Using these two sides of the triangle, we can find the distance from B to D.
Applying the Pythagorean theorem, we have:
Distance^2 = 8^2 - 7^2
Distance^2 = 64 - 49
Distance^2 = 15
Distance = √15
Distance of D from B ≈ 3.87 km
To find the distance of point D from point C, we can treat points B, C, and D as forming a right-angled triangle.
b) Distance of D from C:
Using the same method as above, we have:
Distance^2 = 9^2 - 8^2
Distance^2 = 81 - 64
Distance^2 = 17
Distance = √17
Distance of D from C ≈ 4.12 km
Hence, the distance of D from B is approximately 3.87 km, and the distance of D from C is approximately 4.12 km.
Draw the diagram and review the Pythagorean Theorem.
DB^2 = 8^2 - 7^2
DC^2 = 9^2 + DB^2