Work out the area of this trapezium
6cm
10cm
8cm
17cm

  (81√15)/4 ≈ 78.43 cm²

Step-by-step explanation:

It can help to draw a diagram.

The base angle at the side of length 6 can be found using the Law of Cosines. Its opposite side is of length 8, and the two adjacent sides are of length 6 and 7. So, we have ...

  8² = 6² +7² -2·6·7·cos(α)

  cos(α) = (36+49-64)/(84) = 1/4

  α = arccos(1/4) ≈ 75.522°

Then the height of the trapezium is ...

  6·sin(75.522°) ≈ 5.80948 . . . . cm

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The area of the trapezium is given by the formula ...

  A = 1/2(b1 +b2)h

  A = (1/2)(17 +10)(5.80948) . . . . cm²

  A ≈ 78.43 cm²

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Of course, sin(arccos(1/4)) = (√15)/4, so the exact formulation of the area is ...

  A = (81√15)/4 cm²

I'm not sure exactly but my best guess would be 1/4

Answer: question no. 1 -a. yes, the sides are in the ratio 2 : 5.question no. 2 -b. 59°question no. 3 -c. angles x and y are remote interior angles.question no. 4 -d. 90°, 45° and 45°question no. 5 -b. mn and qrstep-by-step explanation: question no. 1 -rs : xu = 3 : 7.5 = 2 : 5rt : x? = 6.4 : 16 = 2 : 5st : u? = 6 : 15 = 2 : 5as the ratios of all sides are equivalent, the triangles must be congruent and the sides must be proportional.therefore, the answer is: a. yes, the sides are in the ratio 2 : 5.question no. 2 -in the second diagram, angle b is the only letter not shown. therefore, the number must be equivalent to angle b.if we substitute the value of angle b as being 22° in the first diagram, we can now use the angle sum of a triangle to obtain the value of angle a.angle sum of a triangle = 180°therefore: let angle a = aa + 99 + 22 = 180a = 180 - 99 - 22a = 59°therefore, the answer is: b. 59°question no. 3 -angles x and y don't share the vertex of the triangle with the exterior angle w. remote interior angles don't share a vertex of a triangle with an exterior angle. this means that angles x and y are remote interior angles.therefore, the answer is: c. angles x and y are remote interior angles.question no. 4 -the sum of all interior angles of a triangle must be equivalent to 180°.90°, 45° and 45° are the only set of angle measures that are equivalent to 180°.therefore, the answer is: d. 90°, 45° and 45°question no. 5 -the only pair of sides stated in the options which are corresponding are mn and qr.therefore, the answer is: b. mn and qr.