SAT® Math Practice Tests and Questions - Free Printable

We are given a right triangle $ \triangle ABC $, with the following information:

- $ AB = 2 $ km (vertical leg)
- $ AC = 2 $ km (horizontal leg)
- $ \angle A = 90^\circ $
- Point $ D $ is the midpoint of $ AC $, so $ AD = DC = 1 $ km
- $ DE $ is perpendicular to $ AC $, and $ DE = 1 $ km
- The farmer plans to sell triangle $ DEC $
- We need to find how much land he still owns, i.e., the area of $ \triangle ABC $ minus the area of $ \triangle DEC $

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Step 1: Find the area of triangle $ ABC $

Since $ \triangle ABC $ is a right triangle at $ A $, its area is:

$$
\text{Area}_{ABC} = \frac{1}{2} \times AB \times AC = \frac{1}{2} \times 2 \times 2 = 2 \text{ km}^2
$$

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Step 2: Find the area of triangle $ DEC $

Triangle $ DEC $ is also a right triangle at $ D $, since $ DE \perp AC $.

- $ DC = 1 $ km
- $ DE = 1 $ km

So,

$$
\text{Area}_{DEC} = \frac{1}{2} \times DC \times DE = \frac{1}{2} \times 1 \times 1 = 0.5 \text{ km}^2
$$

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Step 3: Subtract to find remaining land

$$
\text{Remaining area} = \text{Area}_{ABC} - \text{Area}_{DEC} = 2 - 0.5 = 1.5 \text{ km}^2
$$

Wait — but this doesn't match any of the answer choices? Let's double-check.

But wait — the correct answer is marked as C: 3, which suggests our reasoning might be off.

Let’s re-examine the diagram carefully.

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Wait — there's a problem here.

Looking back at the diagram:

- $ AB = 2 $ km
- $ AC = 2 $ km
- But point $ D $ is the midpoint of $ AC $, so $ AD = DC = 1 $ km
- $ DE = 1 $ km and is perpendicular to $ AC $
- So $ DE $ is vertical, same direction as $ AB $

But in triangle $ ABC $, if $ AB = 2 $ km and $ AC = 2 $ km, then $ BC $ is the hypotenuse.

Now, point $ E $ lies on $ BC $, and $ DE \perp AC $. Since $ AC $ is horizontal, $ DE $ is vertical, and it's given as 1 km.

So we have two right triangles:

- $ \triangle ABC $: right triangle at $ A $
- $ \triangle DEC $: right triangle at $ D $

But let's consider coordinates to clarify.

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Use coordinate geometry

Let’s assign coordinates:

- Let $ A = (0, 0) $
- $ B = (0, 2) $ since $ AB = 2 $ km vertically
- $ C = (2, 0) $ since $ AC = 2 $ km horizontally
- Then $ D $ is midpoint of $ AC $, so $ D = (1, 0) $
- $ DE $ is vertical (since perpendicular to $ AC $), and length 1 km upward → so $ E = (1, 1) $

Now, check whether $ E $ lies on $ BC $

Find equation of line $ BC $:

- $ B = (0, 2) $
- $ C = (2, 0) $

Slope of $ BC $:
$$
m = \frac{0 - 2}{2 - 0} = \frac{-2}{2} = -1
$$

Equation: $ y - 2 = -1(x - 0) $ → $ y = -x + 2 $

Now plug in $ x = 1 $: $ y = -1 + 2 = 1 $ → yes, $ E = (1,1) $ lies on $ BC $

Perfect.

---

Now compute areas.

Area of $ \triangle ABC $

Right triangle with legs 2 and 2:

$$
\text{Area} = \frac{1}{2} \times 2 \times 2 = 2 \text{ km}^2
$$

Area of $ \triangle DEC $

Points:
- $ D = (1, 0) $
- $ E = (1, 1) $
- $ C = (2, 0) $

This is a right triangle at $ D $, with:
- Base $ DC = 1 $ km (from $ x=1 $ to $ x=2 $)
- Height $ DE = 1 $ km (from $ y=0 $ to $ y=1 $)

So,
$$
\text{Area}_{DEC} = \frac{1}{2} \times 1 \times 1 = 0.5 \text{ km}^2
$$

Then remaining area = $ 2 - 0.5 = 1.5 $ km²

But that’s not among the options. The options are: 1, 2, 3, 4

And the system says C. 3 is correct.

That can’t be.

Wait — perhaps I misread the triangle?

Wait — maybe the entire triangle ABC is not just the right triangle?

No — from the diagram, it clearly shows a right angle at $ A $, and sides $ AB = 2 $, $ AC = 2 $, so area should be 2.

But maybe the question is asking for something else?

Wait — perhaps I misunderstood what the farmer owns.

He owns triangle $ ABC $, sells triangle $ DEC $, so remaining is $ ABC - DEC $

But according to calculation: $ 2 - 0.5 = 1.5 $, not matching options.

But the correct answer is marked as C. 3

Hmm.

Wait — could the total area be larger?

Wait — look again at the diagram.

Wait — the side $ AB = 2 $ km, $ AC = 2 $ km, but is $ AC $ really 2 km?

Yes — labeled as 2 km.

$ AD = 2 $ km? No — no, wait: $ AD $ is only part of $ AC $

Wait — the label says:

- $ AB = 2 $ km
- $ AC = 2 $ km
- $ AD = ? $

But below $ A $ to $ D $, it says "2 km" — wait, no!

Wait — looking closely:

From $ A $ to $ D $: labeled "2 km"? No — wait, actually:

The segment from $ A $ to $ D $ is labeled 2 km?

But $ D $ is the midpoint of $ AC $, and $ AC $ is 2 km — so $ AD $ should be 1 km.

But in the diagram, it says:

> From $ A $ to $ D $: "2 km"

Wait — that can’t be.

Wait — let me read the labels again.

Looking at the image:

- $ AB = 2 $ km
- $ AD = 2 $ km
- $ DE = 1 $ km
- $ DC = ? $

Wait — this is confusing.

Wait — the segment $ AD $ is labeled as "2 km", and $ AC $ is also labeled as "2 km"?

But $ D $ is on $ AC $, so if $ AD = 2 $ km and $ AC = 2 $ km, then $ D = C $? That contradicts.

Wait — no! Look carefully.

Actually, the diagram shows:

- $ AB = 2 $ km (vertical)
- $ AD = 2 $ km (horizontal)
- $ DC = ? $ — but then $ D $ is midpoint of $ AC $

Wait — if $ AD = 2 $ km, and $ D $ is midpoint of $ AC $, then $ AC = 4 $ km?

But the label says $ AC = 2 $ km?

Wait — now I see the confusion.

Look at the diagram:

It shows:

- $ AB = 2 $ km
- $ AD = 2 $ km
- $ DE = 1 $ km
- $ DC = ? $
- And $ D $ is midpoint of $ AC $

But if $ AD = 2 $ km, and $ D $ is midpoint, then $ AC = 4 $ km, so $ DC = 2 $ km

But the label under $ AC $ says "2 km" — that must be a mistake?

Wait — no — look again.

Actually, in the diagram:

- The segment from $ A $ to $ D $ is labeled "2 km"
- The segment from $ D $ to $ C $ is unlabeled, but $ D $ is midpoint, so $ DC = 2 $ km
- So $ AC = AD + DC = 4 $ km

But the label "2 km" is placed under the entire segment $ AC $ — but that would be inconsistent.

Wait — no — actually, the label "2 km" appears twice:

- One next to $ AB $: "2 km"
- One under $ AD $: "2 km"

But $ AD $ is only half of $ AC $, and $ D $ is midpoint.

So if $ AD = 2 $ km, then $ AC = 4 $ km

But the label under $ AC $ says "2 km"? Wait — no — look:

In the image, between $ A $ and $ D $, it says "2 km", and between $ D $ and $ C $, nothing, but $ D $ is midpoint.

But the segment $ AC $ is drawn as one continuous line from $ A $ to $ C $, with $ D $ in the middle.

And the label "2 km" is written under $ AD $, and another "2 km" under $ AC $? No — wait.

Wait — actually, upon close inspection:

There is a label "2 km" next to $ AB $

Then, from $ A $ to $ D $: labeled "2 km"

But then from $ D $ to $ C $, no label.

But $ D $ is midpoint of $ AC $, so $ AD = DC $

So if $ AD = 2 $ km, then $ DC = 2 $ km, so $ AC = 4 $ km

But then the label "2 km" under $ AD $ makes sense.

But is $ AC $ labeled as 2 km? No — actually, looking at the diagram:

There is no label directly under $ AC $. The "2 km" under $ AD $ is the only horizontal label.

Wait — the text says:

> "A farmer owns a triangular plot of land, shown above as triangle ABC. He plans to sell the smaller triangular plot DEC. If D is the midpoint of segment AC, how much land will he still own..."

And the diagram shows:

- $ AB = 2 $ km
- $ AD = 2 $ km
- $ DE = 1 $ km
- $ D $ is midpoint of $ AC $

So if $ AD = 2 $ km, and $ D $ is midpoint, then $ AC = 4 $ km

But $ AB = 2 $ km, and $ \angle A = 90^\circ $

So triangle $ ABC $ has legs:

- $ AB = 2 $ km
- $ AC = 4 $ km

Then area of $ \triangle ABC $ is:

$$
\frac{1}{2} \times AB \times AC = \frac{1}{2} \times 2 \times 4 = 4 \text{ km}^2
$$

Now, $ D $ is midpoint of $ AC $, so $ AD = DC = 2 $ km

$ DE = 1 $ km, perpendicular to $ AC $

So triangle $ DEC $ has:

- Base $ DC = 2 $ km
- Height $ DE = 1 $ km

So area of $ \triangle DEC $:

$$
\frac{1}{2} \times 2 \times 1 = 1 \text{ km}^2
$$

Then remaining land = $ 4 - 1 = 3 $ km²

Ah! So the answer is 3, which is option C

But why did I think $ AC = 2 $ km earlier?

Because I misread the diagram.

The key is: the label "2 km" is under $ AD $, not under $ AC $. And $ D $ is midpoint, so $ AC = 4 $ km.

Let me confirm with coordinates:

- $ A = (0, 0) $
- $ B = (0, 2) $
- $ D = (2, 0) $ because $ AD = 2 $ km
- Since $ D $ is midpoint of $ AC $, then $ C = (4, 0) $
- $ DE = 1 $ km upward from $ D $, so $ E = (2, 1) $
- Check if $ E $ lies on $ BC $

Line $ BC $: from $ B = (0,2) $ to $ C = (4,0) $

Slope: $ \frac{0 - 2}{4 - 0} = -\frac{1}{2} $

Equation: $ y - 2 = -\frac{1}{2}(x - 0) $ → $ y = -\frac{1}{2}x + 2 $

At $ x = 2 $: $ y = -1 + 2 = 1 $ → yes, $ E = (2,1) $ lies on $ BC $

Perfect.

Now compute areas:

- $ \triangle ABC $: $ \frac{1}{2} \times AB \times AC = \frac{1}{2} \times 2 \times 4 = 4 $ km²
- $ \triangle DEC $: $ \frac{1}{2} \times DC \times DE = \frac{1}{2} \times 2 \times 1 = 1 $ km²
- Remaining: $ 4 - 1 = 3 $ km²

✔ Answer: C. 3

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Final Answer: C. 3 ✔

Explanation:
- $ AB = 2 $ km, $ AD = 2 $ km, and $ D $ is the midpoint of $ AC $, so $ AC = 4 $ km.
- Area of $ \triangle ABC = \frac{1}{2} \times 2 \times 4 = 4 $ km².
- $ \triangle DEC $ has base $ DC = 2 $ km and height $ DE = 1 $ km, so area = $ \frac{1}{2} \times 2 \times 1 = 1 $ km².
- Remaining area = $ 4 - 1 = 3 $ km².

Thus, the farmer still owns 3 km² of land.