ISEE Upper Level

Segments & Lengths

Segment arithmetic, midpoints, bisectors, and line segment relationships — 1D geometry without angle measurement

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Imagine you have a giant, 10-foot-long submarine sandwich. 🥪 If you want to share it with your best friend, you'll probably chop it right down the middle. In geometry, that sandwich is a line segment—a straight line with a clear beginning and end. And that exact middle spot? That's called the midpoint!

On the ISEE, you'll see lots of questions about segments and lengths. The test makers love to give you a long segment chopped into smaller pieces and ask you to figure out the missing lengths. It's basically a fun math puzzle! If you know the whole sandwich is 10 feet long, and your friend eats 4 feet of it, you know you have exactly 6 feet left. This is called the Segment Addition Postulate (which is just a super fancy way of saying 'small piece + small piece = whole thing').

Sometimes, the ISEE Quantitative Reasoning section will ask you to compare two different segments. Just remember: a bisector is like the knife that cuts a segment exactly in half, creating two perfectly equal pieces. 📏 If a question mentions a midpoint or a bisector, you instantly know you have two matching halves. Keep your eyes peeled for those magic keywords, always draw a quick picture on your scratch paper, and you'll be a segment-solving superstar on test day!

Key Formula

If point

B

is between

A

and

C

, then

A B + B C = A C

. If

M

is the exact midpoint of

A C

, then

A M = M C = \frac{A C}{2}

Practice Questions

4 practice questions for ISEE Upper Level

Q1 Hard

Points A, B, C, and D are collinear and arranged in that order on a line. M is the midpoint of

A B

and N is the midpoint of

C D

. If

A C = 32

and

B D = 38

, what is the length of

M N

30

35

32

38

Show Solution

Let $A B = x$ , $B C = y$ , and $C D = z$ .

Since A, B, C, D are collinear in that order, we have:
$A C = A B + B C = x + y$
$B D = B C + C D = y + z$
We are given $A C = 32$ , so $x + y = 32$ .
We are given $B D = 38$ , so $y + z = 38$ .
M is the midpoint of $A B$ , so $A M = M B = x /2$ .
N is the midpoint of $C D$ , so $C N = N D = z /2$ .
We need to find the length of $M N$ .
$M N = M B + B C + C N$
Substitute the expressions for $M B$ , $B C$ , and $C N$ :
$M N = x /2 + y + z /2$
$M N = (x + z) /2 + y$
From $x + y = 32$ , we can write $x = 32 - y$ .
From $y + z = 38$ , we can write $z = 38 - y$ .
Now substitute these expressions for $x$ and $z$ into the equation for $M N$ :
$M N = ((32 - y) + (38 - y)) /2 + y$
$M N = (70 - 2 y) /2 + y$
$M N = 35 - y + y$
$M N = 35$
Therefore, the length of $M N$ is $35$ .

Answer: B

Q2 Hard

A line segment

P Q

is divided by points

R

and

S

such that P, R, S, Q are in order on the line. The lengths of

P R

R S

, and

S Q

are in the ratio 2 : 3 : 4. If the distance from point

P

to the midpoint of segment

S Q

63

units, what is the total length of segment

P Q

72

81

90

99

Show Solution

Let the common ratio be $k$ .

Then $P R = 2 k$ , $R S = 3 k$ , and $S Q = 4 k$ .
Let $M$ be the midpoint of segment $S Q$ .
Since $S Q = 4 k$ , the length $S M = M Q = S Q /2 = 4 k /2 = 2 k$ .
The distance from point $P$ to the midpoint $M$ of $S Q$ is given by $P M$ .
$P M = P R + R S + S M$
Substitute the expressions in terms of $k$ :
$P M = 2 k + 3 k + 2 k$
$P M = 7 k$
We are given that $P M = 63$ units.
So, $7 k = 63$ .
Divide by $7$ to find $k$ :
$k = \frac{63}{7}$
$k = 9$
The total length of segment $P Q$ is $P Q = P R + R S + S Q$ .
$P Q = 2 k + 3 k + 4 k$
$P Q = 9 k$
Now substitute the value of $k$ :
$P Q = 9 \times 9$
$P Q = 81$
The total length of segment $P Q$ is $81$ units.

Answer: B

Q3 Hard

Points A, B, C, D are collinear such that

B

is between

A

and

C

, and

C

is between

B

and

D

. Given

A B = 7

C D = 9

, and

A C + B D = 30

. What is the length of

A D

21

23

25

26

Show Solution

The arrangement of the points is $A - B - C - D$ .

We are given:
1. $A B = 7$
2. $C D = 9$
3. $A C + B D = 30$

From the collinear arrangement, we can express $A C$ and $B D$ in terms of $A B$ , $B C$ , and $C D$ :
$A C = A B + B C$
$B D = B C + C D$
Substitute the given values for $A B$ and $C D$ :
$A C = 7 + B C$
$B D = B C + 9$
Now, substitute these expressions into the equation $A C + B D = 30$ :
$(7 + B C) + (B C + 9) = 30$
Combine like terms:
$7 + 9 + B C + B C = 30$
$16 + 2 B C = 30$
Subtract $16$ from both sides:
$2 B C = 30 - 16$
$2 B C = 14$
Divide by $2$ to find $B C$ :
$B C = \frac{14}{2}$
$B C = 7$
Finally, we need to find the length of $A D$ .
$A D = A B + B C + C D$
Substitute the known values:
$A D = 7 + 7 + 9$
$A D = 14 + 9$
$A D = 23$
The length of $A D$ is $23$ .

Answer: B

Q4 Hard

Points K, L, M, N are collinear in that order on a line. Point

P

bisects segment

L M

. If

K L = x + 3

L P = 2 x - 1

, and

M N = x + 7

, and the total length of segment

K M

24

units, what is the length of segment

K N

33.8

34.6

35.2

35.6

Show Solution

The points are arranged as $K - L - P - M - N$ because $P$ bisects $L M$ .

We are given:
1. $K L = x + 3$
2. $L P = 2 x - 1$
3. $M N = x + 7$
4. $K M = 24$

Since $P$ bisects $L M$ , it means $L P = P M$ .
So, $P M = 2 x - 1$ .
The length of $L M$ is $L P + P M$ :
$L M = (2 x - 1) + (2 x - 1)$
$L M = 4 x - 2$
Now, consider the length $K M$ .
$K M = K L + L M$
Substitute the expressions for $K L$ and $L M$ :
$K M = (x + 3) + (4 x - 2)$
We are given $K M = 24$ :
$24 = (x + 3) + (4 x - 2)$
Combine like terms:
$24 = 5 x + 1$
Subtract $1$ from both sides:
$23 = 5 x$
Divide by $5$ to find the value of $x$ :
$x = \frac{23}{5}$
$x = 4.6$
Finally, we need to find the length of $K N$ .
$K N = K L + L M + M N$
Substitute the expressions for each segment:
$K N = (x + 3) + (4 x - 2) + (x + 7)$
Combine like terms:
$K N = (x + 4 x + x) + (3 - 2 + 7)$
$K N = 6 x + 8$
Now substitute the value of $x = 4.6$ :
$K N = 6 (4.6) + 8$
$K N = 27.6 + 8$
$K N = 35.6$
The length of segment $K N$ is $35.6$ units.

Answer: D

Tips & Strategies

Always draw a picture! ✏️ The ISEE often describes segments using only words. Drawing a quick line and labeling the points will instantly make the math easier to see.
Pay close attention to what the question is actually asking. If you solve for $x$ , double-check if the test wants the value of $x$ or the length of the entire segment. They love to put the value of $x$ as a trap answer!
Look for magic words like 'midpoint' and 'bisect'. These are secret codes telling you that two pieces are exactly equal.

Common Mistakes

Watch out for assuming a point is a midpoint just because it's drawn in the middle. On the ISEE, unless the problem specifically says 'midpoint', 'bisects', or shows equal tick marks, you cannot assume the two halves are equal!
Don't forget to answer every question! There is NO penalty for guessing on the ISEE. If you're stuck on a segment puzzle, take your best guess and move on.

Frequently Asked Questions

Do I need a ruler for the ISEE?

Nope! In fact, you aren't allowed to bring a ruler. The shapes on the test are usually 'not drawn to scale', which means you have to rely on the numbers and formulas, not your eyes, to solve the problem.

What does it mean if points are 'collinear'?

Collinear is just a fancy geometry word that means 'on the same line'. If the ISEE says points A, B, and C are collinear, you can draw one straight line right through all of them!

What's the difference between Quantitative Reasoning and Math Achievement?

Great question! Quantitative Reasoning has less reading and uses more logic, including Column A vs. Column B questions. Math Achievement tests what you've learned in school, so you'll see more standard math problems and word problems there.

What if I get a fraction when solving for a segment length?

Don't panic! Segment lengths can absolutely be fractions or decimals. If a segment is $5$ units long, its midpoint will split it into two equal pieces that are each $2 \frac{1}{2}$ or $\frac{5}{2}$ units long.

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