Mathematics
Grade 10
15 min
Choose numbers with a particular product
Choose numbers with a particular product
What you'll learn
- Identify at least three different pairs of whole numbers that multiply to a target product between 20 and 100.
- Solve for missing factors in multiplication equations with a product between 20 and 100, demonstrating understanding of the relationship between multiplication and division.
- Explain the strategy used to find number pairs with a given product, using mathematical vocabulary like 'factor,' 'product,' and 'multiple'.
- Apply knowledge of multiplication facts to efficiently determine if a given number is a factor of a target product between 20 and 100.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify the correct theorem for intersecting chords, secants, and tangents.
Apply the Intersecting Chords Theorem to find unknown segment lengths.
Use the Secant-Secant and Tangent-Secant theorems to solve for unknown lengths.
Calculate the constant product, known as the 'Power of a Point', for a given geometric configuration.
Choose different pairs of integer lengths for segments that satisfy a particular product.
Set up and solve algebraic equations derived from circle theorems.
What if you could find a hidden multiplication rule inside every circle? 🧐 Let's explore the secret constant product that connects intersecting lines!
In this tutorial, we will uncover the powerful relationship between the segments formed by intersecting chords...
2
Key Concepts & Vocabulary
TermDefinitionExample
ChordA line segment whose two endpoints lie on the circle.If you draw a line from one side of a pizza crust to the other without passing through the center, that's a chord.
SecantA line that passes through a circle, intersecting it at two distinct points.Imagine a laser beam passing through a planet; the path of the beam is a secant line.
TangentA line that touches a circle at exactly one point, called the point of tangency.A ruler resting on the side of a ball is tangent to the ball.
Power of a PointA constant value calculated by multiplying the lengths of segments formed by a line passing through a given point and a circle. This product is the same for any line passing through that point.If two chords intersect, and the segments of one are 2 and 6, the Power o...
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Core Formulas
Intersecting Chords Theorem
a \cdot b = c \cdot d
When two chords intersect inside a circle, the product of the segments of one chord (a and b) is equal to the product of the segments of the other chord (c and d).
Secant-Secant Theorem
a \cdot (a+b) = c \cdot (c+d)
When two secants are drawn from an external point, the product of the length of the whole first secant (a+b) and its external part (a) equals the product of the length of the whole second secant (c+d) and its external part (c).
Tangent-Secant Theorem
t^2 = a \cdot (a+b)
When a tangent and a secant are drawn from an external point, the square of the tangent's length (t) equals the product of the length of the whole secant (a+b) and its external part (a).
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Challenging
A secant from an external point P has an external segment of length x and an internal segment of length x+2. A tangent from the same point P has length 2√6. Find the value of x.
A.1
B.2
C.3
D.9
Challenging
From an external point P, two secants and one tangent are drawn. The first secant has an external part of 4 and an internal part of 12. The second secant has an external part of 8. What is the length of the tangent segment from P?
A.8
B.64
C.16
D.4√3
Challenging
If the Power of a Point P with respect to a circle is exactly 0, where must point P be located?
A.Inside the circle, but not at the center
B.At the center of the circle
C.Outside the circle
D.On the circle
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What grade level is "Choose numbers with a particular product"?
Choose numbers with a particular product is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Choose numbers with a particular product?
You'll be able to: Identify at least three different pairs of whole numbers that multiply to a target product between 20 and 100; Solve for missing factors in multiplication equations with a product between 20 and 100, demonstrating understanding….
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How many practice questions are included with Choose numbers with a particular product?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.