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Samuel W.

Mathematics Student, Tutor for 5 years

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Trigonometry

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Question:

A right angled triangle has hypotenuse of length 5cm, and one of the sides is length 4cm. What is the length of the remaining side?

Samuel W.

Answer:

As we know, a triangle is a three-sided shape. The question has given us the length of two of the sides and wants us to work out the remaining side. The other important piece of information is that the triangle is right-angled. This means that Pythagoras' Theorem works for this triangle. Pythagoras' Theorem says that the square of the hypotenuse of a right angled triangle (the side opposite the right angle) is equal to the sum of the squares of the other two sides. We know that the hypotenuse has length 5cm. So first we need to square this. $$5^2 = 25$$ So we know that when we square the other two sides and add them up, we should get 25. Another way of writing this is... $$a^2 + b^2 = 25$$ where $$a$$ and $$b$$ represent the lengths of the other two sides. We know the length of one of the other sides is 4cm. Let's call this side a, giving us... $$4^2 + b^2 = 25$$ $$16 + b^2 = 25$$ $$b^2 = 9$$ $$b = 3$$ So the other side (which we have called b) has length 3cm. So 3cm is the solution to the question.

Geometry

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Question:

Which has greater area? A parallelogram with base length 10cm and perpendicular height 6cm, or a circle of diameter 8cm.

Samuel W.

Answer:

The question requires you to work out the area of the two shapes individually and then compare your answers to see which is greater. Firstly, to calculate the area of the parallelogram you must recall the formula 'area = base x perpendicular height'. The perpendicular height is the maximum length of a line drawn at a right angle (90 degrees) from the base. This formula makes sense because you can imagine cutting off one 'overhanging' end of the parallelogram and attaching it to the other side to form a rectangle. This is why the formula is similar to the area of a rectangle (base x height). Using this formula the area of the parallelogram = 10 x 6 = $${60cm}^2$$. For the area of a circle, you need to remember the formula ' area = $$\pi$$ x $${radius}^2$$ '. There are many ways of remembering this very important formula including rhymes and songs. $$\pi$$ is a fixed number and can be found on a calculator so there is no need to work this out. However to get the radius of the circle, you need to remember that radius = diameter divided by 2. This is because the diameter is the distance from one side of the circle to the other opposite side, while radius is the distance from the centre to the edge. Using this, we can work out the radius of the circle = 8 / 2 = 4. Putting this into the formula for area we get, area of circle = $$\pi$$ x 42 = $$\pi$$ x 16 = $${50.27cm}^2$$ (to two decimal places). We can now see that the area of the parallelogram ($${60cm}^2$$) is greater than the area of the circle ($${50.27cm}^2$$) so the answer is the parallelogram.

Algebra

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Question:

Solve the simultaneous equations: 2x + y = 7 y - 1 = x

Samuel W.

Answer:

There are several methods for solving simultaneous equations, but a fast and fairly simple way is by substitution, which I will demonstrate here. First we need to write one of the unknowns (x or y) in terms of the other. Here, we can see from the second equation that x = y - 1. The next step is to substitute this expression for x into the other equation. This gives us... 2x + y = 7 2 (y -1) + y = 7 2y - 2 + y = 7 3y - 2 = 7 3y = 9 y = 3 So we now know that y = 3. We can now simply substitute this back into the first equation to find the value of x. So... x = y - 1 x = 3 - 1 x = 2 So the solution to the question is x =2 and y = 3

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