You Can't Do That! Quiz / Test - Geometry Problems | Math

Quiz Answer Key and Fun Facts

1. We start with an infamous Pythagoras' Theorem question. The hardworking student Alicia takes one look at it and says: "this is a 3-4-5 triangle", a specific type of triangle that satisfies Pythagoras' Theorem, as 3^2 + 4^2 = 5^2. However, the missing side length is not 5 cm long. What's the problem?

Answer: There's no right angle

Pythagoras' Theorem only works under the condition that the triangle has a right angle. If you've got a 120� angle in a triangle then there can't be a right (90�) angle, as the angles need to add up to 180�.

If you draw two lines of lengths 3 cm and 4 cm with a 120� angle between them, notice that there's only one way to turn it into a triangle. If you compared your drawing to someone else's, they might have a different orientation or mirror image to you, but their missing side length will be the same as yours.

So, the question "what's the missing length?" does have a unique answer. We would call all triangles with two lengths and the angle between them in common "congruent" to each other.

To find the length (without measuring), we'd need the cosine rule: a^2 = b^2 + c^2 - 2bc*cosA. If the missing length is x then we would find that x^2 = 3^2 + 4^2 - 2*3*4*cos120 and eventually reach x = 6.08 cm (to 2 decimal places)

We can then ask: what's stopping me from using the cosine rule on a right-angled triangle, rather than Pythagoras' Theorem? And the answer is, "nothing". In the case that A is a right angle, cosA = 0 so 2bc*cosA = 0 and the cosine rule reduces to a^2 = b^2 + c^2, which is Pythagoras' Theorem (perhaps with slightly different variable naming to how you know it).

2. The corresponding angles rule is nicknamed "the F rule" (with friends "the C rule" and "the Z rule") because of the shape you need to look for in a diagram to find corresponding angles. The eager Alicia immediately writes down: "angle x� is 120� because corresponding angles are equal". However, here our "F rule" cannot be used to conclude this. Why not?

Answer: The horizontal lines are not parallel

The "F", "C" and "Z" rules apply no matter how the diagram is orientated: you might be looking at a mirror image or upside-down "F", but the "corresponding angles" rule will still hold.

However, in the case of "F" and "Z" angles, we do need the appropriate sides to be parallel: here, one line is exactly horizontal but the top line is inclined upwards. Unlike in the letter "F", the other line does not need to be exactly perpendicular (here, vertical), otherwise the two angles would always be right angles of 90�.

3. "Angles on a straight line add up to 180�" is what Alicia says when shown this image. We've been given one angle, 100�, so the other marked angle, x, must be 80�. What's the problem?

Answer: A missing angle is not included

There's an angle between the 100� and the x�. If we called that angle y� then we would know that 100�+x�+y� = 180�, but without knowing something about y�, we are at a loss on how to calculate x�. We can, however, say that it must be no more than 80�.

4. "This one is pi*4*4 = 16pi cm^2", Alicia says, when asked to find the area of the given circle. But it's not. What common mistake has she made?

Answer: 4 cm is the diameter, not the radius

The radius of a circle is the distance from the center to the circumference (exterior). Why is there only one possible distance, when we could draw any number of different lines from center to circumference? Well, that's the defining property of a circle: every point on the circumference is equally far from the center.

The diameter, then, joins two points on the circumference while passing through the center. It is double the length of the radius, because it is in fact two radii joined together (at a 180� angle). Here, 4 cm is the diameter, so each radius is 2 cm.

We can then use the formula to find that the area is pi*2*2 = 4pi cm^2.

The formula for circumference would use the diameter, pi*d. Or we could just as well use the radius, with 2*pi*r. For a circle, "circumference" and "perimeter" mean the same thing.

5. What is the area of this triangle? Alicia recalls "half times the base times the height" as the rule, or "1/2bh" symbolically. So what's the problem with the proposed calculation of the area, 1/2*4*7 = 14 cm^2?

Answer: The height is not 7 cm

The vertical height of the triangle is not 7 cm: the side labelled is the slant height. To find the vertical height (which is not a pretty number in this case), you could draw the vertical height in as an unknown side length, and use Pythagoras' Theorem with the information we have. This assumes that the triangle is isosceles (has two sides the same length), as it looks to be if the diagram is drawn to scale.

cm^2 are the correct units because area is a two-dimensional quantity.

Confusion is sometimes had between the formula for an area of a rectangle, base times height (bh), and that for a triangle (1/2bh). However, if you draw a triangle and rectangle with the same base length and vertical height, it should be clear that the triangle has a smaller area. It's, furthermore, possible to split the rectangle up into two triangles, showing that the triangle area is half of the rectangle area.

6. "Angles in a quadrilateral (four-sided polygon) add up to 360�. So the angles in this shape should sum to 360�: a�+b�+c�+d�+e� = 360�", Alicia thinks. Her friend Benjamin says, "you can't do that!" What's the problem?

Answer: The shape is not a quadrilateral

The shape has four sides, but one side is curved. A "polygon" is a 2D shape where each side is straight.

The four angles in a four-sided polygon do indeed add up to 360�, but we have five "angles" labelled on this non-polygon. The values c�, d� and e� labelled on the diagram are not really angles at all, as they are connected to parts of a curved line. An angle joins two straight lines.

7. "Opposite angles in a cyclic quadrilateral add up to 180�", says Alicia. But Bob takes a look at the angle x� and knows that it can't be 80� - "it's clearly bigger than a right angle, which would be 90�". Which of them "can't do that!", and why not?

Answer: Alicia, as x is not on the circumference of the circle

Eyeballing an angle can be misleading if the diagram is not drawn to scale, but Bob's instinct is a good one, as the angle looks obtuse: more than a right angle (90�), but less than 180�.

A cyclic quadrilateral is a four-sided shape where each vertex (corner) is touching the circumference (exterior) of the circle. The shape shown is four-sided, but not cyclic, as the center is not on the circumference.

All quadrilaterals have four angles that sum to 360�, but cyclic quadrilaterals have the more specific property that each pair of opposite angles sum to 180�.

8. Bob takes a punt at the next question: "it's a triangle, so the angles add up to 180�, and we already have two marked angles of 90� and 30�, so that makes x� = 60�". Alicia says, "you can't do this... and something about this question isn't right". What isn't right?

Answer: The shape is not a triangle

The marked "angle" is not actually an angle of this shape, because the little kink at the top of the shape means that it has four sides, so it is not a triangle but a quadrilateral.

As such, it has four angles, not three, and those angles would add up to 360�.

We could actually still measure the marked angle, x�, but the labels alone don't give us enough information to work out what it is, other than that it must be less than 60�, because of the way the kink in the triangle is at an incline.

9. Alicia is trying to find the gradient of this curve, and she recites the rule "the gradient is the change in y over the change in x". She plots two points on the curve, A(0,0) and B(2,4). According to Alicia, the change in y is 4-0 = 4, and the change in x is 2-0 = 2. Then, 4/2 = 2, so the gradient of the curve is 2. What's the problem?

Answer: The curve does not have a constant gradient

The gradient, or slope, measures the rate of change: as you move one square to the right, does the curve move up or down? And by how much?

Only a straight line has the same gradient everywhere - on a straight line, it doesn't matter which two points you pick, or which way around you subtract the co-ordinates (so long as you do it the same way around for both).

However, more complicated graphs, such as this curve, the quadratic y = x^2, do not have a constant gradient everywhere. The gradient has to be negative when the curve is decreasing (when x is negative) and positive when the curve is increasing (when x is positive), for a start.

The field of calculus is dedicated to measuring the gradient of curves at each particular point: the gradient at (0, 0) happens to be 0, and the gradient at (2, 4) is 4. The gradient at (x, x^2) is 2x. This can be found by a calculation known as "differentiation".

10. Lastly, let's help Alicia and Bob, who are confused about this question. Alicia is insisting that we can use "SOHCAHTOA" to find the missing side length, x. SOHCAHTOA helps us select the right trig value, either "sine", "cosine" or "tangent" depending on which two sides out of "hypotenuse", "opposite" and "adjacent" we are looking at in the question. "It must be 8*sin(30�) = 4 cm long", Alicia evaluates. Bob does not agree. What is the problem?

Answer: SOHCAHTOA only applies in a right-angled triangle

SOHCAHTOA relies on the quite surprising fact that no matter how you draw a right-angled triangle with three fixed angles, the ratio of each side is always in a fixed proportion. For instance, in a triangle with a right angle and two 45� angles, the shorter two sides are the same length, and the hypotenuse is about 1.414... times longer.

Here, we have no (labelled) right angle. In fact, there is no way to calculate the length x, because the information given isn't enough to draw a unique triangle! Imagine tilting the 8 cm side length up and down - the lengths of the other two sides of the triangle would have to change so that the 30� angle remains fixed.

Lots of different values of x are possible, so we would need another angle or side length to begin any sort of calculations. Just one more piece of information will make the triangle unique (up to congruence), so then we can use the sine rule or cosine rule to find the length x.

Source: Author AdamM7

This quiz was reviewed by FunTrivia editor rossian before going online.
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You Can't Do That! Trivia Quiz

Geometry Problems

In a math classroom, many teachers never ask "when can this rule be used?", so students often try to use formulas and theorems when they do not apply. In these high school geometry problems, let's follow Alicia and Bob as we see why "You Can't Do That!"

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